Sharp Geometric Inequalities for the General <Emphasis Type="Italic">p</Emphasis>-Affine Capacity

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.