Note on the Isoperimetric Profile of a Convex Body

Summary

A solution to the relative isoperimetric problem (or partitioning problem) is a subset E of a given set Ω ⊂ ℝⁿ (the container) with prescribed volume |E| =V and minimal area $\int_\Omega {\left| {D_{XE} } \right|\, = \,A(V)}$ of the interface. Improving a result due to Sternberg & Zumbrun [8], we obtain that if Ω is convex then $A{(V)^{{\frac{n}{{n - 1}}}}}$ is a concave function of V. As consequence we deduce that the isoperimetric ratio of E is no worse than that of the half-ball contained in ℍ = ℝ^n-1 X (0, ∞), and that the mean curvature of the minimizer is bounded a priori.