Sharp bounds for the anisotropic $p$-capacity of Euclidean compact sets

We prove various sharp bounds for the anisotropic $p$-capacity ${\mathrm{C}\mathrm{a}\mathrm{p}}_{F,p}(K)$ ($1<p<n$) of compact sets $K$ in the Euclidean space ${\mathbb{R}}^n$ ($n\ge 3$). For example, using the inverse anisotropic mean curvature flow (IAMCF), we get an upper bound of Szeg"{o} type (1931) for ${\mathrm{C}\mathrm{a}\mathrm{p}}_{F,p}(K)$ when $\mathrm{\partial }K$ is a smooth, star-shaped and $F$-mean convex hypersurface in ${\mathbb{R}}^n$ ($n\ge 3$). Moreover, for such a surface $\mathrm{\partial }K$ in ${\mathbb{R}}^3$, by introducing the anisotropic Hawking mass and studying its monotonicity property along IAMCF, we obtain an upper bound of Bray–Miao type (2008) for ${\mathrm{C}\mathrm{a}\mathrm{p}}_{F,p}(K)$.