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Constantin Vernicos

Publications and source records attributed to Constantin Vernicos.

17 recordsLinked to original sources
arXiv2023

Volume growth of Funk geometry and the flags of polytopes

We consider the Holmes--Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santal\'o point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon.

math.MG↗
arXiv2018

Flag-approximability of convex bodies and volume growth of Hilbert geometries

We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both quantities are maximized when the convex body is a Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume.

math.MG↗
arXiv2016

Approximability of convex bodies and volume entropy in Hilbert geometry

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).

math.MG↗
arXiv2014

On the Hilbert Geometry of Convex Polytopes

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.

math.MG↗
arXiv2012

Asymptotic volume in Hilbert Geometries

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

math.MG↗
arXiv2011

On the Hilbert Geometry of products

We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that amenability of a product is equivalent to the amenability of each terms.

math.DG↗
arXiv2011

Feuilletage de Hirsch, mesure harmonique, et g-mesure

Les feuilletages de Hirsch sont des feuilletages par surfaces de variétés compactes fermées de dimension 3, dont la dynamique transverse est celle d'un endomorphisme du cercle de degré strictement supérieur à 1. Le but de cette note est de construire, à partir d'une g-mesure associée à un tel endomorphisme, une mesure harmonique sur le feuilletage de Hirsch correspondant, au sens de Lucy Garnett. Ceci nous permet de donner des exemples de métriques riemanniennes sur le fibré tangent du feuilletage de Hirsch, lisses le long des feuilles et continues transversalement, pour lesquelles il existe plusieurs mesures harmoniques. De tels exemples montrent que le résultat d'unique ergodicité obtenu par le premier auteur et Victor Kleptsyn pour les feuilletages transversalement conformes [GAFA, 2007, Vol. 17, No 4, 1043-1105] n'est valable que lorsque la métrique riemannienne est Höldérienne transversalement, mais pas juste continue.

math.DS↗
arXiv2008

Volume entropy of Hilbert Geometries

It is shown that the volume entropy of a Hilbert geometry associated to an $n$-dimensional convex body of class $C^{1,1}$ equals $n-1$. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed. In the case $n=2$, and without any assumption on the boundary, it is shown that the entropy is bounded above by $\frac{2}{3-d} \leq 1$, where $d$ is the Minkowski dimension of the extremal set of $K$. An example of a plane Hilbert geometry with entropy strictly between 0 and 1 is constructed.

math.DG↗
arXiv2007

Spectral Radius and Amenability in Hilbert Geometries

We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a Fölner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the spectrum of the Hyperbolic geometry of the same dimension. Horoballs, from a purely metric point of view, and their relation with the bottom of the spectrum in Hilbert geometries are briefly studied.

math.DG↗
arXiv2006

Les géométries de Hilbert sont à géométrie locale bornée

We prove that the Hilbert geometry of a convex domain in ${\mathbb R}^n$ has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of ${\mathbb R}^n$. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We also give a counter exemple in dimension three which shows that the reciprocal is not true for non plane Hilbert geometries.

math.DG↗
arXiv2003

L'aire des triangles ideaux en geometrie de Hilbert

In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the convex, which characterizes the triangle domains in the plane. Furthermore, under specific geometric assumptions, we obtain an upper bound, which depends on the convex domain.

math.DG↗
arXiv2003

The Length of Harmonic Forms on a Compact Riemannian Manifold

We study $n$ dimensional Riemanniann manifolds with harmonic forms of constant length and first Betti number equal to $n-1$ showing that they are 2-steps nilmanifolds with some special metrics. We also characterise, in terms of properties on the product of harmonic forms, the left invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

math.DG↗
arXiv2003

The macroscopical sound of tori

Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes to infinity, and characterise the flat tori using the tools of homogenisation our conclusion being that "Macroscopically, one can hear the shape of a flat torus".

math.DG↗
arXiv2002

The macroscopic spectrum of nilmanifolds with an emphasis on the heisenberg groups

Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian for the dirichlet problem on them. We describe the asymptotic behaviour of the spectrum when the radius of these balls goes to infinity. Furthermore we show that the first macroscopic eigenvalue is bounded from above, by an uniform constant for the three dimensional heisenberg group, and by a constant depending on the Albanese's torus for the other nilmanifolds. We also show that the Heisenberg groups belong to a family of nilmanifolds, where the equality characterizes some pseudo left invariant metrics.

math.DG↗