Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems

Let $Ω\subset \mathbb{R}^N$ be a bounded convex domain and let $H_Ω\leq 0$ denote its variational mean curvature, in the sense of Barozzi, Gonzalez and Tamanini [PAMS, 1987]. We prove that $-H_Ω$ is harmonic convex, that is $1/H_Ω$ is a convex function in $Ω$, and that the exponent $-1$ is optimal i