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.