284 research outputs found
Affine images of isotropic measures
Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies. © 2015, International Press of Boston, Inc. All rights reserved
The log-Brunn-Minkowski inequality
For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are "equivalent" in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies. © 2012 Elsevier Ltd
Estimates for measures of sections of convex bodies
A estimate in the hyperplane problem with arbitrary measures has
recently been proved in \cite{K3}. In this note we present analogs of this
result for sections of lower dimensions and in the complex case. We deduce
these inequalities from stability in comparison problems for different
generalizations of intersection bodies
Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies
Let K \subset R^N be a convex body containing the origin. A measurable set G
\subset R^N with positive Lebesgue measure is said to be uniformly K-dense if,
for any fixed r > 0, the measure of G \cap (x + rK) is constant when x varies
on the boundary of G (here, x + rK denotes a translation of a dilation of K).
We first prove that G must always be strictly convex and at least C1,1-regular;
also, if K is centrally symmetric, K must be strictly convex, C1,1-regular and
such that K = G - G up to homotheties; this implies in turn that G must be
C2,1- regular. Then for N = 2, we prove that G is uniformly K-dense if and only
if K and G are homothetic to the same ellipse. This result was already proven
by Amar, Berrone and Gianni in [3]. However, our proof removes their regularity
assumptions on K and G and, more importantly, it is susceptible to be
generalized to higher dimension since, by the use of Minkowski's inequality and
an affine inequality, avoids the delicate computations of the higher-order
terms in the Taylor expansion near r = 0 for the measure of G\cap(x+rK) (needed
in [3])
Valuations on lattice polytopes
This survey is on classification results for valuations defined on lattice polytopes that intertwine the special linear group over the integers. The basic real valued valuations, the coefficients of the Ehrhart polynomial, are introduced and their characterization by Betke and Kneser is discussed. More recent results include classification theorems for vector and convex body valued valuations. © Springer International Publishing AG 2017
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