$L_p$ geominimal surface areas and their inequalities

TL;DR

This paper introduces a new extension of the $L_p$ geominimal surface area for all $p$ in a specific range, establishing fundamental properties and inequalities that generalize previous concepts in convex geometry.

Contribution

It extends the $L_p$ geominimal surface area to a broader range of p, providing new properties, inequalities, and comparisons in convex geometric analysis.

Findings

Proved affine isoperimetric inequality for the new $L_p$ geominimal surface area.

Established Santaló style inequality and cyclic inequalities.

Demonstrated monotonicity and comparison with $p$-surface area.

Abstract

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santal\'{o} style inequality. Cyclic inequalities are established to obtain the monotonicity of the $L_p$ geominimal surface areas. Comparison between the $L_p$ geominimal surface area and the $p$-surface area is also provided.