General Measure Extensions of Projection Bodies

Abstract

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for ${\text{vol}}_n^{n-1}(K){\text{vol}}_n({\Pi }^{\circ }K),$ where $\Pi K$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $\Pi$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

48 pages. In the Proceedings of the London Mathematical Society

Cited by in corpus (4)

• Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures
• Weighted Minkowski's Existence Theorem and Projection Bodies
• Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
• Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications