On the sine polarity and the L-p-sine Blaschke-Santalo inequality

摘要

This paper is dedicated to study the sine version of polar bodies and establish the L-p-sine Blaschke-Santalo inequality for the L-p-sine centroid body. The L-p-sine centroid body Lambda K-p for a star body K subset of R-n is a convex body based on the L-p-sine transform, and its associated Blaschke-Santalo inequality provides an upper bound for the volume of Lambda(p)degrees K, the polar body of Lambda K-p, in terms of the volume of K. Thus, this inequality can be viewed as the "sine cousin" of the L-p Blaschke-Santalo inequality established by Lutwak and Zhang. As p -> infinity, the limit of Lambda(p)degrees K becomes the sine polar body K-lozenge and hence the L-p-sine Blaschke-Santalo inequality reduces to the sine Blaschke-Santalo inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies Cen, which consists of all originsymmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in C-e(n) are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santalo inequality are settled. (c) 2022 Elsevier Inc. All rights reserved.