Extension of the first mixed volume to nonconvex sets

2016/07/26 by Tsukerman, Emmanuel
#52A39 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1607.07802

Abstract

We study the first mixed volume for nonconvex sets and apply the results to limits of discrete isoperimetric problems. Let M,N ⊂ ℝ^d. Define D_N (M)=lim_{ε\downarrow 0} \frac|M+εN|-|M|ε whenever the limit exists. Our main result states that for a compact domain M ⊂ ℝ^d with piecewise C¹ boundary and bounded N ⊂ ℝ^d, D_N(M)=D_conv(N)(M) and D_N(M)=∫_{bd M} h_N(u_M(x)) d H^d-1(x).

Related

• Sets with High Volume and Low Perimeter
• Existence of isoperimetric sets with densities "converging from below" in ℝ^N
• Monotonicity of the optimal perimeter in isoperimetric problems on ℤ^k × ℕ^d
• Integer Subsets with High Volume and Low Perimeter
• Nonexistence of isoperimetric sets in spaces of positive curvature