On the $m$th order $p$-affine capacity

TL;DR

This paper introduces a new mathematical framework for the $m$th order $p$-affine capacity, establishing its properties and inequalities, extending classical affine geometry concepts to higher orders and dimensions.

Contribution

It develops the theory of the $m$th order $p$-affine capacity, including definitions, fundamental properties, and inequalities, advancing affine geometric analysis.

Findings

Defined the $m$th order $p$-affine capacity and proved its invariance properties.

Established inequalities relating the capacity to volume and surface areas.

Compared the $m$th order $p$-affine capacity with other geometric measures.

Abstract

Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area.