Extension of the first mixed volume to nonconvex sets

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 $D_N(M)=\int_{\text{bd }M} h_N(u_M(x)) \, d \mathcal{H}^{d-1}(x)$.