Minimal boundaries enclosing a given volume

Total variation and Cheeger sets in Gauss space

“…The uniqueness and convexity of the Cheeger set inside bounded convex subsets of R n was proved in [15] under the assumption that the set is uniformly convex and of class C 2 , and extended in [1] to the general case. If the ambient set is convex, the C 1,1 -regularity of Cheeger sets is a consequence of the results in [21,22,27]. Moreover, a Cheeger set can be characterized in terms of the mean curvature of its boundary; the sum of the principal curvatures being bounded by the Cheeger constant (see [19,9,24,3] for n = 2 and [2,1] for the general case).…”

Area minimizing sets subject to a volume constraint in a convex set

“…Next, we recall some facts concerning area minimizing sets with a volume constraint. The main result of [GMT1] is that if E is area minimizing with a volume constraint, then…”

“…However, with H Ω denoting the union of all largest balls in Ω, if |H Ω | ≤ v < |Ω|, then E is unique. In addition for such v we show that perimeter minimizers E are nested as a function of v. In general for nonconvex Ω one can expect neither uniqueness or nestedness as indicated by examples in [GMT1]. The nestedness of perimeter minimizers allows one to rearrange level sets of functions to create test functions useful in studying minimizers to certain variational problems.…”

Some special aspects related to the 1-Laplace operator

“…For a Cheeger set C of Ω we know that the surface ∂C ∩ Ω has constant mean curvature h(Ω), cf. Gonzalez et al [14,Theorem 2]. In the case Ω ⊂ R 2 this readily implies that ∂C ∩ Ω consists of circular arcs.…”