New $L_p$ Affine Isoperimetric Inequalities

TL;DR

This paper introduces new $L_p$ affine isoperimetric inequalities valid for all $p$ in the range [-infinity, 1), including a duality formula linking the affine surface areas of a convex body and its polar, expanding the theoretical framework.

Contribution

The paper establishes new $L_p$ affine isoperimetric inequalities for all $p$ in [-infinity, 1) and proves a duality formula connecting affine surface areas of convex bodies and their polars.

Findings

Established $L_p$ affine isoperimetric inequalities for all $p$ in [-infinity, 1)

Proved a duality formula relating affine surface areas of convex bodies and their polars

Extended the theoretical understanding of affine isoperimetric inequalities

Abstract

We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$.