# Interface free energy/attained

The result is quite general and is stated in full generality. Suppose that $$K\subset {\mathbf R}^d\ ,$$ $$d\geq 2\ ,$$ has a nonzero volume $$|K|$$ and that $${\mathbf n}(s)$$ denotes the outward unit normal to its boundary $$\partial K$$ at $$s\ .$$ Let $$W_\tau\subset {\mathbf R}^d$$ be a convex body and $$\tau$$ its support function, $$\tau=\sup\{\langle {\mathbf x}|{\mathbf y}\rangle \colon {\mathbf y}\in W_\tau\}\ ,$$ so that $$W_\tau=\{{\mathbf x}\colon \langle {\mathbf x}|{\mathbf n}\rangle\,,\;\forall {\mathbf n}\}\ .$$ Define ${\mathcal F}(\partial K):=\int_{\partial K}\tau({\mathbf n}(s))\,d{\mathcal H}^{d-1}(s)\,.$ ($$d{\mathcal H}^{d-1}$$ is the $$(d-1)$$-Hausdorff measure in $${\mathbf R}^d\ .$$) The following isoperimetric inequality gives the solution to the variational problem of minimizing the functional $${\mathcal F}(\partial K)$$ among a class of subsets with fixed volume. Roughly speaking, the subsets which can be considered are those subsets for which the functional $${\mathcal F}$$ is well-defined. Then $\tag{1} {\mathcal F}(\partial K)\geq d|W_\tau|^{1/d} |K|^{(d-1)/d}\,.$

Equality holds if and only iff $$K$$ and $$W_\tau$$ differ up to dilation and translation. If one considers only convex sets $$K\ ,$$ then there is a simple proof of (1). The first step is to observe that ${\mathcal F}(\partial K)=\lim_{\varepsilon\downarrow 0} {|K+\varepsilon W_\tau|-|K|\over \varepsilon}.$ Here $$\varepsilon W_\tau=\{\varepsilon x:\,x\in W_\tau\}$$ and $$K+\varepsilon W_\tau=\bigcup_{x\in K}\,(x+\varepsilon W_\tau)\ .$$ This formula is easily proved for polytopes. Then the result follows by applying Brunn-Minkowski inequality to $$K+\varepsilon W_\tau\ ,$$ $|K+\varepsilon W_\tau|\geq \Big(|K|^{1/d}+|\varepsilon W_\tau|^{1/d}\Big)^d,$ with equality if and only if $$K$$ and $$W_\tau$$ differ up to dilation and translation.