# Sharp geometric inequalities for the general $p$-affine capacity

## TL;DR

This paper introduces the general p-affine capacity, explores its properties, and establishes sharp geometric inequalities that extend classical affine isoperimetric and isocapacitary inequalities.

## Contribution

It proposes a new concept of p-affine capacity, analyzes its properties, and derives sharp inequalities linking it to classical geometric quantities.

## Key findings

- The general p-affine capacity is affine invariant and monotone.
- Sharp inequalities relate the p-affine capacity to volume and surface area.
- Extensions of classical isoperimetric inequalities are established.

## Abstract

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities.