Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors w i . Restricted and functional versions of the inequality will also be considered.1 k ,

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“…On the other hand, in the same work [7], the following generalisation of (1.5) was obtained: if S ⊆ [n] has cardinality |S| = d and (S 1 , . .…”

Section: Introduction and Notationmentioning

confidence: 92%

Rogers–Shephard and local Loomis–Whitney type inequalities

et al. 2019

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“…Moreover, a generalization of (6.5) also appeared in [Xi]. In particular, [BGL,Theorem 1.5] provides an isomorphic version of inequality (1.3)…”

Section: Weakly Decomposable Bodiesmentioning

confidence: 99%

Wulff shapes and a characterization of simplices via a Bezout type inequality

Saroglou

Soprunov

Zvavitch

2019

Advances in Mathematics

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Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumesWe show that the above inequality characterizes simplices, i.e. if K is a convex body satisfying the inequality for all convex bodies L1, . . . , Ln ⊂ R n , then K must be an n -dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest.In addition, we study the Bezout inequality for mixed volumes introduced in [SZ]. We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in [SZ] characterizes weakly indecomposable convex bodies.2010 Mathematics Subject Classification. Primary 52A20, 52A39, 52A40, 52B11.

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“…Let us also mention that the Bollobás-Thomason inequality plays a key role in the recent work [8] of S. Brazitikos, A. Giannopoulos and the author that provides local versions of the Loomis-Whitney inequality for coordinate projections of convex bodies; see also [1] for further results in this direction. It is conceivable that one might exploit the dual inequality of Theorem 1.2 to obtain analogous local inequalities for sections.…”

Section: Introductionmentioning

confidence: 99%