A characterization of Blaschke addition

We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

show abstract

“…A characterization of the Blaschke sum was recently obtain by Gardner, Parapatits and Schuster (see [5]). Remark 2.2.…”

Section: Addition Operations On Convex Bodiesmentioning

confidence: 96%

Characterizing addition of convex sets by polynomiality of volume and by the homothety operation

Milman

Rotem

2015

Commun. Contemp. Math.

View full text Add to dashboard Cite

show abstract

“…The operation of the L p -harmonic radial addition and L p -dual Minkowski, Brunn-Minkwski inequalities are the basic concept and inequalities in the L p -dual Brunn-Minkowski theory. The latest information and important results of this theory can be referred to [32,37,39,40,[47][48][49][50][51] and the references therein. For a systematic investigation on the concepts of the addition for convex body and star body, we refer the reader to [26,48,50].…”

Section: Introductionmentioning

confidence: 99%

The Dual Orlicz–Aleksandrov–Fenchel Inequality

Zhao

2020

Mathematics

View full text Add to dashboard Cite

In this paper, the classical dual mixed volume of star bodies V˜(K1,⋯,Kn) and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it Orlicz multiple dual mixed volume. We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of Lp-dual multiple mixed volumes and Lp-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.

show abstract

“…Projection bodies and intersection bodies play a critical role in the solution of Shephard's problem, respectively the Busemann-Petty problem. We refer the reader to [16], [17], [33], [5], [4], [9], [10], [12], [13], [14], [18], [19], [20], [23], [22]. The projection body operator and intersection body operator are continuous and GL(n) contravariant valuations, see [16], [17].…”

Section: Introductionmentioning

confidence: 99%