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In this article, we propose the notion of the general $p$-affine capacity andprove some basic properties for the general $p$-affine capacity, such as affineinvariance and monotonicity. The newly proposed general $p$-affine capacity iscompared 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 $p$-affinecapacity are obtained. These inequalities extend and strengthen many well-known(affine) isoperimetric and (affine) isocapacitary inequalities.

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