Littérature scientifique sur le sujet « Conjecture de Viterbo »

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct, for every integer $n\geqslant 2$, a tight contact form with systolic ratio arbitrarily close to $n$ and with suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.

Abstract We apply the billiard technique to deduce some results on Viterbo’s conjectured inequality between the volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo’s conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo’s conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.

We set out to demonstrate that the Rényi entropies are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the Rényi’s postulates lead to Pap’s g-calculus where the functions carrying out the domain transformation are Rényi’s information function and its inverse. In its turn, Pap’s g-calculus under Rényi’s information function transforms the set of positive reals into a family of semirings where “standard” product has been transformed into sum and “standard” sum into a power-emphasized sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead to idempotent analysis and tropical algebra as well as to less exotic structures. We conjecture that this is one of the reasons why tropical algebra procedures, like the Viterbi algorithm of dynamic programming, morphological processing, or neural networks are so successful in computational intelligence applications. But also, why there seem to exist so many computational intelligence procedures to deal with “information” at large.

AbstractWe prove that the cylindrical capacity of a dynamically convex domain in ${\mathbb{R}}^{4}$ agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in ${\mathbb{R}}^{4}$ which are sufficiently C3 close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.

AbstractWe prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

AbstractIn this paper, we relate Viterbo’s conjecture from symplectic geometry to Minkowski versions of worm problems which are inspired by the well-known Moser worm problem from geometry. For the special case of Lagrangian products this relation provides a connection to systolic Minkowski billiard inequalities and Mahler’s conjecture from convex geometry. Moreover, we use the above relation in order to transfer Viterbo’s conjecture to a conjecture for the longstanding open Wetzel problem which also can be expressed as a systolic Euclidean billiard inequality and for which we discuss an algorithmic approach in order to find a new lower bound. Finally, we point out that the above mentioned relation between Viterbo’s conjecture and Minkowski worm problems has a structural similarity to the known relationship between Bellmann’s lost-in-a-forest problem and the original Moser worm problem.

La convexité joue un rôle particulier en géométrie symplectique, pourtant ce n'est pas une notion invariante par symplectomorphisme. Dans un article fondateur, Hofer, Wysocki et Zehnder ont montré que tout domaine fortement convexe est dynamiquement convexe, une notion, qui elle, est invariante par symplectomorphisme. Depuis plus de vingt ans, l'existence ou non de domaines dynamiquement convexes qui ne sont pas symplectomorphes à un convexe est restée une question ouverte. Récemment, Chaidez et Edtmair ont répondu à cette question en dimension 4. Ils ont établi un critère "quantitatif" de convexité symplectique puis ont construit des domaines dynamiquement convexes qui ne vérifient pas ce critère. Dans cette thèse, nous utilisons ce critère pour construire de nouveaux exemples de tels domaines en dimension 4, qui ont la propriété additionnelle d'être torique. De plus, nous estimons les constantes intervenant dans ce critère. Ce travail en collaboration avec Jean Gutt et Jun Zhang a été ensuite utilisé par Chaidez et Edtmair pour résoudre la question initiale en toute dimension. Dans un second temps, en collaboration avec Jean Gutt, Vinicius G.B.Ramos et Jun Zhang, nous étudions la distance des domaines dynamiquement convexes aux domaines symplectiquement convexes. Nous montrons qu'en dimension 4, celle-ci est arbitrairement grande aux yeux d'un analogue symplectique de la distance de Banach-Mazur. Au passage, nous reprouvons de manière indépendante l'existence de domaines dynamiquement convexes non symplectiquement convexes en dimension 4
Convexity plays a special role in symplectic geometry, but it is not a notion that is invariant by symplectomorphism. In a seminal work, Hofer, Wysocki and Zehnder showed that any strongly convex domain is dynamically convex, a notion that is invariant by symplectomorphism. For more than twenty years, the existence or not of dynamically convex domains that are not symplectomorphic to a convex domain has remained an open question. Recently, Chaidez and Edtmair answered this question in dimension 4. They established a "quantitative" criterion of symplectic convexity and constructed dynamically convex domains that do not satisfy this criterion. In this thesis, we use this criterion to construct new examples of such domains in dimension 4, which have the additional property of being toric. Moreover, we estimate the constants involved in this criterion. This work in collaboration with Jean Gutt and Jun Zhang was later used by Chaidez and Edtmair to solve the initial question in all dimensions. Furthermore, in collaboration with Jean Gutt, Vinicius G.B.Ramos and Jun Zhang, we study the distance from dynamically convex domains to symplectically convex domains. We show that in dimension 4, this distance is arbitrarily large with respect to a symplectic analogue of the Banach-Mazur distance. Additionally, we independently reprove the existence of dynamically convex domains that are not symplectically convex in dimension 4