On the sine polarity and the $L_p$-sine Blaschke-Santal\'{o} inequality

TL;DR

This paper introduces the $L_p$-sine Blaschke-Santaló inequality, a sine version of the classical inequality, and explores the associated convex bodies and their properties, including equality conditions.

Contribution

It establishes the $L_p$-sine Blaschke-Santaló inequality and develops a new class of convex bodies generated by cylinders, expanding the understanding of sine polarity in convex geometry.

Findings

Established the $L_p$-sine Blaschke-Santaló inequality.

Developed the concept of convex bodies generated by cylinders.

Determined the equality conditions for the sine Blaschke-Santaló inequality.

Abstract

This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santal\'{o} inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $\Lambda_p K$ for a star body $K\subset\mathbb{R}^n$ is a convex body based on the $L_p$-sine transform, and its associated Blaschke-Santal\'{o} inequality provides an upper bound for the volume of $\Lambda_p^{\circ}K$, the polar body of $\Lambda_p K$, in terms of the volume of $K$. Thus, this inequality can be viewed as the "sine cousin" of the $L_p$ Blaschke-Santal\'{o} inequality established by Lutwak and Zhang. As $p\rightarrow \infty$, the limit of $\Lambda_p^{\circ} K$ becomes the sine polar body $K^{\diamond}$ and hence the $L_p$-sine Blaschke-Santal\'{o} inequality reduces to the sine Blaschke-Santal\'{o} inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies $\mathcal{C}_{e}^n$, which consists of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in $\mathcal{C}_{e}^n$ 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-Santal\'{o} inequality are settled.