Journal articles: 'Convex projective geometry'

1

Wienhard, Anna, and Tengren Zhang. "Deforming convex real projective structures." Geometriae Dedicata 192, no. 1 (May 5, 2017): 327–60. http://dx.doi.org/10.1007/s10711-017-0243-z.

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Weisman, Theodore. "Dynamical properties of convex cocompact actions in projective space." Journal of Topology 16, no. 3 (August 2, 2023): 990–1047. http://dx.doi.org/10.1112/topo.12307.

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AbstractWe give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.

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Kapovich, Michael. "Convex projective structures on Gromov–Thurston manifolds." Geometry & Topology 11, no. 3 (September 24, 2007): 1777–830. http://dx.doi.org/10.2140/gt.2007.11.1777.

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Kim, Inkang. "Compactification of Strictly Convex Real Projective Structures." Geometriae Dedicata 113, no. 1 (June 2005): 185–95. http://dx.doi.org/10.1007/s10711-005-0550-7.

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5

Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (November 11, 2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.

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Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

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Hildebrand, Roland. "Optimal Inequalities Between Distances in Convex Projective Domains." Journal of Geometric Analysis 31, no. 11 (May 10, 2021): 11357–85. http://dx.doi.org/10.1007/s12220-021-00684-3.

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Benoist, Yves, and Dominique Hulin. "Cubic differentials and finite volume convex projective surfaces." Geometry & Topology 17, no. 1 (April 8, 2013): 595–620. http://dx.doi.org/10.2140/gt.2013.17.595.

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8

Sioen, M., and S. Verwulgen. "Locally convex approach spaces." Applied General Topology 4, no. 2 (October 1, 2003): 263. http://dx.doi.org/10.4995/agt.2003.2031.

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<p>We continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. We prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. Furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces.</p>

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Bray, Harrison, and David Constantine. "Entropy rigidity for finite volume strictly convex projective manifolds." Geometriae Dedicata 214, no. 1 (May 17, 2021): 543–57. http://dx.doi.org/10.1007/s10711-021-00627-w.

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10

Zimmer, Andrew. "A higher-rank rigidity theorem for convex real projective manifolds." Geometry & Topology 27, no. 7 (September 19, 2023): 2899–936. http://dx.doi.org/10.2140/gt.2023.27.2899.

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Li, Jia Hui, Zhuo Qun Wang, Yi Xi Shen, and Zhong Yuan Dai. "Does any convex quadrilateral have circumscribed ellipses?" Open Mathematics 15, no. 1 (December 16, 2017): 1463–76. http://dx.doi.org/10.1515/math-2017-0117.

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Abstract The past decades have witnessed several well-known beautiful conclusions on four con-cyclic points. With highly promising research value, we profoundly studied circumscribed ellipses of convex quadrilaterals in this paper. Using tools of parallel projective transformation and analytic geometry, we derived several theorems including the proof of the existence of circumscribed ellipses of convex quadrilaterals, the properties of its minimal coverage area, and locus center, respectively. This simple approach lays a solid foundation for its application to three-dimensional situations, which is namely the circumscribed quadric surface of a solid figure and its wide-range utility in construction engineering. Meanwhile, we have a new insight into innate connection of conic sections as well as a taste of beauty and harmony of geometry.

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Ballas, Samuel, Daryl Cooper, and Arielle Leitner. "The moduli space of marked generalized cusps in real projective manifolds." Conformal Geometry and Dynamics of the American Mathematical Society 26, no. 7 (August 17, 2022): 111–64. http://dx.doi.org/10.1090/ecgd/367.

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ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to M × [ 0 , ∞ ) M\times [0,\infty ) where M M is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of π 1 M \pi _1M . It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on M M , and the fiber is a closed cone in the space of cubic differentials. For 3 3 -dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

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13

Adeboye, Ilesanmi, and Daryl Cooper. "The area of convex projective surfaces and Fock–Goncharov coordinates." Journal of Topology and Analysis 12, no. 02 (September 14, 2018): 533–46. http://dx.doi.org/10.1142/s1793525319500560.

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The area of a convex projective surface of genus [Formula: see text] is at least [Formula: see text] where [Formula: see text] is the vector of triangle invariants of Bonahon–Dreyer and [Formula: see text] are the Fock–Goncharov triple ratios.

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Looijenga, Eduard. "Discrete automorphism groups of convex cones of finite type." Compositio Mathematica 150, no. 11 (September 10, 2014): 1939–62. http://dx.doi.org/10.1112/s0010437x14007404.

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AbstractWe investigate subgroups of $\text{SL}(n,\mathbb{Z})$ which preserve an open nondegenerate convex cone in $\mathbb{R}^{n}$ and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on self-dual cones, Weyl groups of certain Kac–Moody algebras, and they do occur in algebraic geometry as the automorphism groups of projective manifolds acting on their ample cones.

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15

Choi, Suhyoung. "Convex decompositions of real projective surfaces. II. Admissible decompositions." Journal of Differential Geometry 40, no. 2 (1994): 239–83. http://dx.doi.org/10.4310/jdg/1214455537.

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16

Choi, Suhyoung. "Convex decompositions of real projective surfaces. I. $\pi$-annuli and convexity." Journal of Differential Geometry 40, no. 1 (1994): 165–208. http://dx.doi.org/10.4310/jdg/1214455291.

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17

Lumiste, Ü. "Betweenness plane geometry and its relationship with convex, linear, and projective plane geometries; 233-251." Proceedings of the Estonian Academy of Sciences. Physics. Mathematics 56, no. 3 (2007): 233. http://dx.doi.org/10.3176/phys.math.2007.3.01.

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18

Ahmad, Shamsatun Nahar, Nor’ Aini Aris, and Azlina Jumadi. "The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials." Scientific Research Journal 16, no. 2 (November 18, 2019): 1. http://dx.doi.org/10.24191/srj.v16i2.5507.

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Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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19

McLean, K. Robin. "Conics and convexity." Mathematical Gazette 98, no. 542 (July 2014): 266–72. http://dx.doi.org/10.1017/s0025557200001303.

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In [1], W. D. Munn proved the following result.Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.

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20

Ahmad, Shamsatun Nahar, Nor’Aini Aris, and Azlina Jumadi. "The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials." Scientific Research Journal 16, no. 2 (November 18, 2019): 1. http://dx.doi.org/10.24191/srj.v16i2.9346.

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Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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21

Heinrich, Markus, and David Gross. "Robustness of Magic and Symmetries of the Stabiliser Polytope." Quantum 3 (April 8, 2019): 132. http://dx.doi.org/10.22331/q-2019-04-08-132.

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We give a new algorithm for computing therobustness of magic- a measure of the utility of quantum states as a computational resource. Our work is motivated by themagic state modelof fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to asmagic statesin this context. Therobustness of magicmeasures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamoussign problem. Surprisingly, the robustness of magic issub- multiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness ofncopies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4n-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem tondimensions. The total run-time of our algorithm, while still exponential inn, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certainsigned quantum weight enumerators. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.

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Izhakian, Zur, Manfred Knebusch, and Louis Rowen. "Supertropical quadratic forms II: Tropical trigonometry and applications." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1633–76. http://dx.doi.org/10.1142/s021819671840012x.

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This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93], where we introduced quadratic forms on a module [Formula: see text] over a supertropical semiring [Formula: see text] and analyzed the set of bilinear companions of a quadratic form [Formula: see text] in case the module [Formula: see text] is free, with fairly complete results if [Formula: see text] is a supersemifield. Given such a companion [Formula: see text], we now classify the pairs of vectors in [Formula: see text] in terms of [Formula: see text] This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy–Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio [Formula: see text] of a pair of anisotropic vectors [Formula: see text] in [Formula: see text]. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93]) of a quadratic form on a free module [Formula: see text] over a field in the simplest cases of interest where [Formula: see text]. In the last part of the paper, we introduce a suitable equivalence relation on [Formula: see text], whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic [Formula: see text] the CS-ratio [Formula: see text] depends only on the rays of [Formula: see text] and [Formula: see text]. We develop essential basics for a kind of convex geometry on the ray-space of [Formula: see text], where the CS-ratios play a major role.

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Chen, Kai-Siang, Gelo Noel M. Tabia, Jebarathinam Chellasamy, Shiladitya Mal, Jun-Yi Wu, and Yeong-Cherng Liang. "Quantum correlations on the no-signaling boundary: self-testing and more." Quantum 7 (July 11, 2023): 1054. http://dx.doi.org/10.22331/q-2023-07-11-1054.

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In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k)-parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.

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Alber, Y. I., R. S. Burachik, and A. N. Iusem. "A proximal point method for nonsmooth convex optimization problems in Banach spaces." Abstract and Applied Analysis 2, no. 1-2 (1997): 97–120. http://dx.doi.org/10.1155/s1085337597000298.

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In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

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25

si, Lin. "Some problems about geometric lattice." MATEC Web of Conferences 173 (2018): 03002. http://dx.doi.org/10.1051/matecconf/201817303002.

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In this paper, the survey about some results of the convex lattice set are given and the invariance of projection problem of convex lattice set is also obtained. And combining a famous result in the graph theory, several conjectures about the convex lattice set are presented.

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26

Zhao, Chang-Jian. "Orlicz mixed projection body." Filomat 37, no. 18 (2023): 5895–907. http://dx.doi.org/10.2298/fil2318895z.

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In the paper, our main aim is to generalize the mixed projection body ?(K1,...,Kn?1) of (n ? 1) convex bodies K1,...,Kn?1 to the Orlicz space. Under the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric operation call it Orlicz mixed projection body ??(K1,...,Kn) of n convex bodies K1,...,Kn. The new affine geometric quantity in special case yields the classical mixed projection body ?(K1,...,Kn?1) and Orlicz projection body ??K of convex body K, respectively. The related concept of Lp-mixed projection body of n convex bodies ?p(K1,...,Kn) is also derived. An Orlicz Alesandrov-Fenchel inequality for the Orlicz mixed projection body is established, which in special case yields a new Lp-projection Alesandrov-Fenchel inequality. As an application, we establish a polar Orlicz Alesandrov-Fenchel inequality for the polar of Orlicz mixed projection body.

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27

Balashov, Maxim V. "Embedding of a homothete in a convex compactum: an algorithm and its convergence." Russian Universities Reports. Mathematics, no. 138 (2022): 143–49. http://dx.doi.org/10.20310/2686-9667-2022-27-138-143-149.

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The problem of covering of a given convex compact set by a homothetic image of another convex compact set with a given homothety center is considered, the coefficient of homothety is calculated. The problem has an old history and is closely related to questions about the Chebyshev center, problems about translates, and other problems of computational geometry. Polyhedral approximation methods and other approximation methods do not work in a space of already moderate dimension (more than 5 on a PC). We propose an approach based on the application of the gradient projection method, which is much less sensitive to dimension than the approximation methods. We select classes of sets for which we can prove the linear convergence rate of the gradient method, i. e. convergence with the rate of a geometric progression with a positive ratio strictly less than 1. These sets must be strongly convex and have, in a certain sense, smoothness of the boundary.

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Talponen, Jarno. "Convex-transitive characterizations of Hilbert spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 3 (May 26, 2009): 633–59. http://dx.doi.org/10.1017/s0308210507000856.

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We investigate real convex-transitive Banach spaces X, which admit a one-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. For example, if uESX is a big point such that there is a bicontractive linear projection P : X → [u] and X* is weak*-locally uniformly rotund, then X is a Hilbert space. The results obtained here are motivated by the well-known Banach—Mazur rotation problem, as well as a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

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Kabluchko, Zakhar, and Hauke Seidel. "Convex cones spanned by regular polytopes." Advances in Geometry 22, no. 2 (April 1, 2022): 245–67. http://dx.doi.org/10.1515/advgeom-2021-0041.

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Abstract We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

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BAROV, STOYU, and JAN J. DIJKSTRA. "ON CLOSED SETS IN HILBERT SPACE WITH CONVEX PROJECTIONS UNDER SOMEWHERE DENSE SETS OF DIRECTIONS." Journal of Topology and Analysis 02, no. 01 (March 2010): 123–43. http://dx.doi.org/10.1142/s1793525310000252.

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Let k be a fixed natural number. In an earlier paper the authors show that if C is a closed and nonconvex set in the Hilbert space ℓ2 such that the closures of the projections onto allk-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of ℓ2. Here this theorem is strengthened significantly by making the much weaker assumption that the set of projection directions is somewhere dense. To show the sharpness of the main theorem we construct "minimal imitations" of closed convex sets in ℓ2. In addition, we show that closed convex sets with an empty geometric interior cannot be imitated by other closed sets.

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Ene, Alina, Huy L. Nguyen, and Adrian Vladu. "Projection-Free Bandit Optimization with Privacy Guarantees." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 8 (May 18, 2021): 7322–30. http://dx.doi.org/10.1609/aaai.v35i8.16899.

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We design differentially private algorithms for the bandit convex optimization problem in the projection-free setting. This setting is important whenever the decision set has a complex geometry, and access to it is done efficiently only through a linear optimization oracle, hence Euclidean projections are unavailable (e.g. matroid polytope, submodular base polytope). This is the first differentially-private algorithm for projection-free bandit optimization, and in fact our bound matches the best known non-private projection-free algorithm and the best known private algorithm, even for the weaker setting when projections are available.

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Li, Xiang Jia, Ning Dai, Wen He Liao, Yu Chun Sun, and Yong Bo Wang. "A Fault-Tolerant Offset Algorithm for Measured Data with Defects." Advanced Materials Research 902 (February 2014): 344–50. http://dx.doi.org/10.4028/www.scientific.net/amr.902.344.

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Offsetting of measured data, as a basic geometric operation, has already been widely used in many areas, like reverse engineering, rapid prototyping and NC machining. However, measured data always carry typical defects like caves and singular points. A fault-tolerant offset method is proposed to create the high quality offset surface of measured data with such defects. Firstly, we generated an expansion sphere model of measured data with the radius equivalent to the offset length. Secondly, using the computational geometry application of convex hull, we acquire the data of outermost enveloping surface of this expansion sphere model. Finally, we use local MLS projection fitting method to wipe out existing defects, and generate the high-quality triangular mesh surface of the offset model. The offset surface generated by this method is suitable for practical engineering application due to its high efficiency and accuracy.

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SEGAL, ALEXANDER, and BOAZ A. SLOMKA. "PROJECTIONS OF LOG-CONCAVE FUNCTIONS." Communications in Contemporary Mathematics 14, no. 05 (August 29, 2012): 1250036. http://dx.doi.org/10.1142/s0219199712500368.

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Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝn containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.

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Stošić, Marko, João Xavier, and Marija Dodig. "Projection on the intersection of convex sets." Linear Algebra and its Applications 509 (November 2016): 191–205. http://dx.doi.org/10.1016/j.laa.2016.07.023.

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Gabeleh, Moosa, Elif Uyanık Ekici, and Manuel De La Sen. "Noncyclic contractions and relatively nonexpansive mappings in strictly convex fuzzy metric spaces." AIMS Mathematics 7, no. 11 (2022): 20230–46. http://dx.doi.org/10.3934/math.20221107.

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<abstract><p>A concept of fuzzy projection operator is introduced and use to investigate the non-emptiness of the fuzzy proximal pairs. We then consider the classes of noncyclic contractions and noncyclic relatively nonexpansive mappings and survey the existence of best proximity pairs for such mappings. In the case that the considered mapping is noncyclic relatively nonexpansive, we need a geometric notion of fuzzy proximal normal structure defined on a nonempty and convex pair in a convex fuzzy metric space. We also prove that every nonempty, compact and convex pair of subsets of a strictly convex fuzzy metric space has the fuzzy proximal normal structure.</p></abstract>

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Gu, Li Zhi. "The Study on the ‘Real Expression’ of Spatial Objects with 2-Dimension and the Real Display Rate." Applied Mechanics and Materials 10-12 (December 2007): 1–5. http://dx.doi.org/10.4028/www.scientific.net/amm.10-12.1.

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It is the ideal goal for the realistic visualization to be accomplished for spatial solids with 2D presentation in digital technology. In the consistently parallel system with the viewing angle and the presenting angle, a technical term, the real presentation rate, is put forward with two indexes, clarity and tri- dimension, taken as an initial criterion of the degree to which the real presentation is reached. By using the projecting geometry, the transform matrix of two-dimensional presentation for the solids was obtained with three steps, for simple and convex body. For more intricate geometric body with concave surfaces, the furry pattern discrimination was introduced, proving that the optimum presentation is that by which the normal iso-axial viewing was adopted. An application has shown that the furry pattern discrimination method worked well, and the trend is the same as the best presentation of an object on 2D among a variety of the solutions.

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Longhas, Paul Ryan A., Alsafat M. Abdul, and Edcon B. Baccay. "The Security Assignment Problem and Its Solution." International Journal of Analysis and Applications 21 (July 17, 2023): 72. http://dx.doi.org/10.28924/2291-8639-21-2023-72.

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In this paper we derived a new method for finding the optimal solution to security assignment problems using projection onto a convex set. This study will help communities find the optimal number of assigned and reserved personnels in designating security officers to an area. This study is applicable as well to CCTV assignment problems. The main goal of this study is to give a new method that can be applied in solving security assignment problems. We used some of the known properties of the convex optimization in proving the properties of the optimal solution, such as the concept of proximity operator, projection onto the convex set, and primal and dual problem. In addition to that, we used some basic knowledge in graph theory to answer our real-life application of this study. The main results of this paper showed that we found instances when the optimal solution to a security problem exists and when the solutions exist, we can determine the answer to the problem explicitly. Also, we proved that there always exists a Pseudo-solution in security assignment problems, and if the solution exists, then the Pseudo-solution will coincide with it. The most important aspect of this paper is the introduction of the application of convex optimization in the security problem.

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Geng, Qing Jia, Xi Juan Guo, and Ya Zhang. "Research on Exact Minkowski Sum Algorithm of Convex Polyhedron Based on Direct Mapping." Advanced Materials Research 225-226 (April 2011): 377–80. http://dx.doi.org/10.4028/www.scientific.net/amr.225-226.377.

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Minkowski sum has become an effective method in collision detection problem, which is a branch of computation geometry. Separated from the previous algorithm based on the traditional Gaussian Map, a new algorithm of computing exact Minkowski sum of convex polyhedron is proposed based on direct mapping method in the paper, and the correctness of direct mapping method is testified. The algorithm mapping the convex polyhedron into the bottom of regular tetrahedron according to the definition of Regular Tetrahedron Mapping and Point Projection, so the problem become form 3D to 2D. Comparing with the previous algorithm, the algorithm posed in the paper establishes mapping from 3D to 2D directivity, and only compute the overlay of one pair of planar subdivision. So, the algorithm’s executing efficiency has been improved in compare with the previous algorithm.

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ASHRAFUL ALAM, MD, and MASUD HASAN. "COMPUTING NICE PROJECTIONS OF CONVEX POLYHEDRA." International Journal of Computational Geometry & Applications 21, no. 01 (February 2011): 71–85. http://dx.doi.org/10.1142/s021819591100355x.

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In an orthogonal projection of a convex polyhedron P, the visibility ratio of a face f (of an edge e) is the ratio of orthogonally projected area of f (length of e) and its actual area (length). In this paper, we give algorithms for nice projections of P such that the minimum visibility ratio among all visible faces (among all visible edges) is maximized.

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Martin, R. R., H. Suzuki, and P. A. C. Varley. "Labeling Engineering Line Drawings Using Depth Reasoning." Journal of Computing and Information Science in Engineering 5, no. 2 (February 21, 2005): 158–67. http://dx.doi.org/10.1115/1.1891045.

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Automatic creation of B-rep models of engineering objects from freehand sketches would benefit designers. One step aims to take a line drawing (with hidden lines removed), and from it deduce an initial three-dimensional (3D) geometric realization of the visible part of the object, including junction and line labels, and depth coordinates. Most methods for producing this frontal geometry use line labeling, which takes little or no account of geometry. Thus, the line labels produced can be unreliable. Our alternative approach inflates a drawing to produce provisional depth coordinates, and from these makes deductions about line labels. Assuming many edges in the drawing are parallel to one of three main orthogonal directions, we first attempt to identify groups of parallel lines aligned with the three major axes of the object. From these, we create and solve a linear system of equations relating vertex coordinates, in the coordinate system of the major axes. We then inflate the drawing in a coordinate system based on the plane of the drawing and depth perpendicular to it. Finally, we use this geometry to identify which lines in the drawing correspond to convex, concave, or occluding edges. We discuss alternative realizations of some of the concepts, how to cope with nonisometric-projection drawings, and how to combine this approach with other labeling techniques to gain the benefits of each. We test our approach using sample drawings chosen to be representative of engineering objects. These highlight difficulties often overlooked in previous papers on line labeling. Our new approach has significant benefits.

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Moretti, Valter, and Davide Pastorello. "Frame functions in finite-dimensional quantum mechanics and its Hamiltonian formulation on complex projective spaces." International Journal of Geometric Methods in Modern Physics 13, no. 02 (January 26, 2016): 1650013. http://dx.doi.org/10.1142/s0219887816500134.

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This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of [Formula: see text] covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the [Formula: see text]-algebra structure of the set of quantum observables in terms of classical-like ones.

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Xia, Hongchuan, and Chunping Zhong. "On complex Berwald metrics which are not conformal changes of complex Minkowski metrics." Advances in Geometry 18, no. 3 (July 26, 2018): 373–84. http://dx.doi.org/10.1515/advgeom-2017-0062.

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AbstractWe investigate a class of complex Finsler metrics on a domain D ⊂ ℂn. Necessary and sufficient conditions for these metrics to be strongly pseudoconvex complex Finsler metrics, or complex Berwald metrics, are given. The complex Berwald metrics constructed in this paper are neither trivial Hermitian metrics nor conformal changes of complex Minkowski metrics. We give a characterization of complex Berwald metrics which are of isotropic holomorphic curvatures, and also give characterizations of complex Finsler metrics of this class to be Kähler Finsler or weakly Kähler Finsler metrics. Moreover, in the strongly convex case, we give characterizations of complex Finsler metrics of this class to be projectively flat Finsler metrics or dually flat Finsler metrics.

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Fang, Zheng, and Juwon Seo. "A Projection Framework for Testing Shape Restrictions That Form Convex Cones." Econometrica 89, no. 5 (2021): 2439–58. http://dx.doi.org/10.3982/ecta17764.

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This paper develops a uniformly valid and asymptotically nonconservative test based on projection for a class of shape restrictions. The key insight we exploit is that these restrictions form convex cones, a simple and yet elegant structure that has been barely harnessed in the literature. Based on a monotonicity property afforded by such a geometric structure, we construct a bootstrap procedure that, unlike many studies in nonstandard settings, dispenses with estimation of local parameter spaces, and the critical values are obtained in a way as simple as computing the test statistic. Moreover, by appealing to strong approximations, our framework accommodates nonparametric regression models as well as distributional/density‐related and structural settings. Since the test entails a tuning parameter (due to the nonstandard nature of the problem), we propose a data‐driven choice and prove its validity. Monte Carlo simulations confirm that our test works well.

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Shkurnikov, Sergey, Ol'ga Bulakaeva, and Vladimir Anisimov. "Complex Spatial Geometry of Curved Sections on High-Speed Railways." Bulletin of scientific research results, no. 2 (June 23, 2022): 164–78. http://dx.doi.org/10.20295/2223-9987-2022-2-164-178.

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Purpose: The paper poses some promising trends of modern projection science development for high-speed railway plan and profile. The main objective of the trends is to critically rethink the provisions of the existing regulatory design base of the projection. The question on inadmissibility of the arrangement of complex spatial geometry sections — the sections of coincidence of a vertical curve in a profile and a passage curve in a plan — is one of such provisions. Methods: In the frames of the conducted research, a theoretical approach to the determination of force impacts at the section of complex spatial geometry is proposed. Results: It is established that force impact value is caused by vertical curve direction (convex or concave), as well as by calculated value of outstanding acceleration in being considered curve, Studies of the influence of sections with complex spatial geometry on the dynamics of high-speed rolling stock motion show that at certain combination of circular and vertical curve radii for motion various speeds, it is possible to minimize negative dynamic effects caused by the presence of complex spatial geometry section. Practical importance: As a result of the work, it was found that at speeds of 350 km/h or more, the superimposition of a vertical curve in the profile and a passage curve in the plan is acceptable without applying additional requirements to the curve parameters. The radius of the circular curve should be increased by 30-60% at speeds of less than 350 km/h.

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Denisov, Sergei, and Vladimir Semenov. "ADAPTIVE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES." Journal of Automation and Information sciences 5 (September 1, 2021): 82–92. http://dx.doi.org/10.34229/1028-0979-2021-5-7.

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Many problems of operations research and mathematical physics can be formulated in the form of variational inequalities. The development and research of algorithms for solving variational inequalities is an actively developing area of applied nonlinear analysis. Note that often nonsmooth optimization problems can be effectively solved if they are reformulated in the form of saddle point problems and algorithms for solving variational inequalities are applied. Recently, there has been progress in the study of algorithms for problems in Banach spaces. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces. A new algorithm for solving variational inequalities in a Banach space is proposed and studied. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

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OLIVEIRA E SILVA, DIOGO, and RENÉ QUILODRÁN. "A comparison principle for convolution measures with applications." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 2 (June 28, 2019): 307–22. http://dx.doi.org/10.1017/s0305004119000197.

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AbstractWe establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

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Ölçmen, Semih M., Stanley E. Jones, and Ryan H. Weiner. "A numerical analysis of projectile nose geometry including sliding friction for penetration into geological targets." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 2 (November 3, 2016): 284–304. http://dx.doi.org/10.1177/0954406216676849.

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Optimal axisymmetric penetrator nose shapes that result in most penetration under the effect of pressure-dependent Coulomb Friction are presented. The nose shapes were determined by dividing a circular cylinder into line segments and iteratively updating the combination of line segment slopes to generate the nose geometries that result in the most penetration. Analysis was conducted for five different penetrator velocities, [Formula: see text], 11 different friction coefficient values, μ, and 10 different nose shape factors, α. Calculations were also made on common nose shapes including a 3/4 power nose, AMNG model, a tangent-ogive nose, and a standard cone to show that the new nose shapes result in the highest penetration. Results show that while most of the optimal nose shapes are convex shapes, the nose shapes become concave at μ = 0.4 for [Formula: see text], μ ≥ 0.6 for [Formula: see text], μ ≥ 0.8 for [Formula: see text] at different α values. The nose shapes are also blunt in general except for [Formula: see text], while μ = 0.1, α = 0.3; μ = 0.2, α = 0.4; μ = 0.5, α = 0.6; for [Formula: see text], while μ = 0.2, α = 0.3; μ = 0.4, α = 0.4; for [Formula: see text], while μ = 0.1, α = 0.2; μ = 0.3, α = 0.3; μ = 0.7, α = 0.4; for [Formula: see text], while μ = 0.1, α = 0.2; μ = 0.4, α = 0.3; μ = 0.5, α = 0.3, where the nose shapes are more close to a cone. Results can be used to select a nose shape that can achieve the desired penetration depth.

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Abbasov, Majid, and Faramoz Aliev. "Modifications of the Charged Balls Method." Open Computer Science 10, no. 1 (February 24, 2020): 30–32. http://dx.doi.org/10.1515/comp-2020-0008.

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AbstractThe Charged Balls Method is based on physical ideas. It allows one to solve problem of finding the minimum distance from a point to a convex closed set with a smooth boundary, finding the minimum distance between two such sets and other problems of computational geometry. This paper proposes several new quick modifications of the method. These modifications are compared with the original Charged Ball Method as well as other optimization methods on a large number of randomly generated model problems.We consider the problem of orthogonal projection of the origin onto an ellipsoid. The main aim is to illustrate the results of numerical experiments of Charged Balls Method and its modifications in comparison with other classical and special methods for the studied problem.

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GOYAL, NAVIN, LUIS RADEMACHER, and SANTOSH VEMPALA. "Query Complexity of Sampling and Small Geometric Partitions." Combinatorics, Probability and Computing 24, no. 5 (October 21, 2014): 733–53. http://dx.doi.org/10.1017/s0963548314000704.

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In this paper we study the following problem.Discrete partitioning problem (DPP). Let$\mathbb{F}_q$Pndenote then-dimensional finite projective space over$\mathbb{F}_q$. For positive integerk⩽n, let {Ai}i= 1Nbe a partition of ($\mathbb{F}_q$Pn)ksuch that:(1)for alli⩽N,Ai= ∏j=1kAji(partition into product sets),(2)for alli⩽N, there is a (k− 1)-dimensional subspaceLi⊆$\mathbb{F}_q$Pnsuch thatAi⊆ (Li)k.What is the minimum value ofNas a function ofq, n, k? We will be mainly interested in the casek=n.DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: when the dimension is low (k=n= 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.

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Zhao, Jiaxin. "Coverage Path Planning with Adaptive Hyperbolic Grid for Step-Stare Imaging System." Drones 8, no. 6 (June 4, 2024): 242. http://dx.doi.org/10.3390/drones8060242.

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Step-stare imaging systems are widely used in aerospace optical remote sensing. In order to achieve fast scanning of the target region, efficient coverage path planning (CPP) is a key challenge. However, traditional CPP methods are mostly designed for fixed cameras and disregard the irregular shape of the sensor’s projection caused by the step-stare rotational motion. To address this problem, this paper proposes an efficient, seamless CPP method with an adaptive hyperbolic grid. First, we convert the coverage problem in Euclidean space to a tiling problem in spherical space. A spherical approximate tiling method based on a zonal isosceles trapezoid is developed to construct a seamless hyperbolic grid. Then, we present a dual-caliper optimization algorithm to further compress the grid and improve the coverage efficiency. Finally, both boustrophedon and branch-and-bound approaches are utilized to generate rotation paths for different scanning scenarios. Experiments were conducted on a custom dataset consisting of 800 diverse geometric regions (including 2 geometry types and 40 samples for 10 groups). The proposed method demonstrates comparable performance of closed-form path length relative to that of a heuristic optimization method while significantly improving real-time capabilities by a minimum factor of 2464. Furthermore, in comparison to traditional rule-based methods, our approach has been shown to reduce the rotational path length by at least 27.29% and 16.71% in circle and convex polygon groups, respectively, indicating a significant improvement in planning efficiency.

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Journal articles on the topic 'Convex projective geometry'

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