Abstract
A new kind of duality between intersection bodies and projection bodies is presented. Furthermore, some inequalities for mixed intersection bodies are established. A geometric inequality is derived between the volumes of star duality of star bodies and their associated mixed intersection integral.
Highlights
Introduction and main resultsIntersection bodies were first explicitly defined and named by Lutwak [1]
A geometric inequality is derived between the volumes of star duality of star bodies and their associated mixed intersection integral
Following Lutwak, the intersection body of order i of a star body is introduced by Zhang [2]
Summary
Some authors including Haberl and Ludwig [5], Kalton and Koldobsky [6], Klain [7, 8], Koldobsky [9], Ludwig [10, 11], and so on have given considerable attention to the intersection bodies and their various properties. The aim of this paper is to establish several inequalities about the star dual version of intersection bodies. We establish the star dual version of the general Busemann intersection inequality. Theorem 1.1 is an analogue of the general Petty projection inequality which was given by Lutwak [12], concerning the polar duality of convex bodies. For two star bodies K and L, let K+ ̆ L denote the radial Blaschke sum of K and L [1]. We establish the dual Brunn-Minkowski inequality for the star duality of mixed intersection bodies concerning the radial Blaschke sum. Theorem 1.3 is an analogue of the general Brunn-Minkowski inequality for the polar duality of mixed projection bodies concerning the Blaschke sum [1].
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