Total positive curvature and the equality case in the relative isoperimetric inequality outside convex domains (Q2683744)

In this paper, the case of equality for the relative isoperimetric inequality outside a convex closed set \(\mathbf{C}\) established by \textit{J. Choe} et al. [Calc. Var. Partial Differ. Equ. 29, No. 4, 421--429 (2007; Zbl 1116.58016)] is settled. In [loc. cit.], it was proved that when \(\mathbf{C}\) has \(C^2\)-boundary, the (bounded, finite-perimiter, and finite-measure) set \(\Omega\) attaining the equality for the relative isoperimetric inequality must be a half ball. In this work, any such \(\Omega\) is shown to be a half ball supported on a facet of the convex set, without the \(C^2\)-assumption on \(\mathbf{C}\). The key to the proof includes analysing the \textit{restricted \(\Lambda\)-minimisers} of the relative perimeter based on smooth approximations of the convex set \(\mathbf{C}\). In particular, one needs to bound the total positive curvature of the boundary of restricted \(\Lambda\)-minimisers outside \(\mathbf{C}\), and to show that restricted \(\Lambda\)-minimisers satisfy a perpendicular contact angle condition ``in a viscosity sense''. In addition, a sharp inequality for the Willmore energy is obtained (Theorem~3.9).