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The log-Brunn-Minkowski inequality
Abstract 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 theExpand
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The logarithmic Minkowski problem
In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of aExpand
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L p Affine Isoperimetric Inequalities
The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (orExpand
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Blaschke-Santaló inequalities
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The affine Sobolev inequality
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Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems
A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally toExpand
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The paper studies convex bodies and star bodies in $\Bbb R^n$ by using Radon transforms on Grassmann manifolds, $p$-cosine transforms on the unit sphere, and convolutions on the rotation group ofExpand
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Sharp Affine LP Sobolev Inequalities
In this paper we prove a sharp affine Lp Sobolev inequality for functions on R. The new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev inequality ofExpand
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Orlicz projection bodies
Abstract Minkowski's projection bodies have evolved into L p projection bodies and their asymmetric analogs. These all turn out to be part of a far larger class of Orlicz projection bodies. TheExpand
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Centered bodies and dual mixed volumes
We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in R" by using dual mixed volumes. One of theExpand
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