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pro vyhledávání: '"Vernicos, P."'
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 t
Externí odkaz:
http://arxiv.org/abs/2306.09268
Autor:
Vernicos, Constantin, Walsh, Cormac
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 m
Externí odkaz:
http://arxiv.org/abs/1809.09471
Autor:
Vernicos, Constantin
Publikováno v:
Pacific J. Math. 287 (2017) 223-256
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
Externí odkaz:
http://arxiv.org/abs/1207.1342
Akademický článek
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Autor:
Vernicos, Constantin
Publikováno v:
In Handbook of Hilbert Geometry, IRMA Lectures in Mathematics and Theoretical Physics Vol. 22, pp. 111-125, 2014
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 simplex
Externí odkaz:
http://arxiv.org/abs/1406.0733
Autor:
Vernicos, Constantin
Publikováno v:
Indiana university Journal of mathematics, 62(5):1431-1441, 2013
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.
Externí odkaz:
http://arxiv.org/abs/1207.1142
Autor:
Vernicos, Constantin
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 p
Externí odkaz:
http://arxiv.org/abs/1109.0187
Autor:
Deroin, Bertrand, Vernicos, Constantin
Publikováno v:
Publicaciones Matem\'aticas del Uruguay, 12:79-85 (2011)
Les feuilletages de Hirsch sont des feuilletages par surfaces de vari\'et\'es compactes ferm\'ees de dimension 3, dont la dynamique transverse est celle d'un endomorphisme du cercle de degr\'e strictement sup\'erieur \`a 1. Le but de cette note est d
Externí odkaz:
http://arxiv.org/abs/1103.0177
Autor:
Vernicos, Constantin
Publikováno v:
Osaka J. Math. 52:215-235 (2014)
We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.
Comment: Extends our previous result with B. Colbois and P. Verovic arXiv:0804.1620v1. Main Theorem i
Comment: Extends our previous result with B. Colbois and P. Verovic arXiv:0804.1620v1. Main Theorem i
Externí odkaz:
http://arxiv.org/abs/0812.1032
Publikováno v:
Pacific J. Math., 245 (2010), 201-225
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 constru
Externí odkaz:
http://arxiv.org/abs/0810.1123