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

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.