Extension of the first mixed volume to nonconvex sets

Abstract

We study the first mixed volume for nonconvex sets and apply the results to limits of discrete isoperimetric problems. Let $M,N\subset {\mathbb{R}}^d$. Define $D_N(M)={\lim }_{\epsilon \downarrow 0}\frac{|M+\epsilon N|-|M|}{\epsilon }$ whenever the limit exists. Our main result states that for a compact domain $M\subset {\mathbb{R}}^d$ with piecewise $C^1$ boundary and bounded $N\subset {\mathbb{R}}^d$, $D_N(M)=D_{\text{conv}(N)}(M)$ and $D_N(M)={\int }_{\text{bd }M}{h}_N(u_M(x))d{\mathcal{H}}^{d-1}(x)$.

7 pages