arXiv subjects

Patrick Verovic

Publications and source records attributed to Patrick Verovic.

8 recordsLinked to original sources

arXiv2022

When is a Minkowski norm strictly sub-convex?

The aim of this paper is to give two complete and simple characterizations of Minkowski norms N on an arbitrary topological real vector space such that the sublevel sets of N are strictly convex. We first show that this property is equivalent to the continuity of N together with the fact that any open chord between two points of the boundary of the sublevel set N^{-1}([0, 1)) lies inside that set (geometric characterization). On the other hand, we prove that this is also the same as saying that N is continuous and that for an arbitrary real number $\alpha$ > 1 the function N^$\alpha$ is strictly convex (analytic characterization).

math.FA↗

arXiv2016

Functions with strictly convex epigraph

The aim of this paper is to provide a complete and simple characterization of functions with domain in a topological real vector space whose epigraph is strictly convex.

math.FA↗

arXiv2012

Laplacian and spectral gap in regular Hilbert geometries

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.

math.DG↗

arXiv2012

Two properties of volume growth entropy in Hilbert geometry

The aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.

math.MG↗

arXiv2009

Hilbert domains quasi-isometric to normed vector spaces

We prove that a Hilbert domain which is quasi-isometric to a normed vector space is actually a convex polytope.

math.MG↗

arXiv2008

Hilbert geometry for convex polygonal domains

We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.

math.MG↗

arXiv2007

Some smooth Finsler deformations of hyperbolic surfaces

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.

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↗