  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.
