Two properties of optimisers for the reverse isoperimetric problem

The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the boundary of the set is of class $C^2$, this amounts to a positive lower bound on the principal curvatures and in this class we prove that there are no $C^2$-maximisers of perimeter with prescribed volume. In addition, we prove that a given possibly non-$C^2$ maximiser has its smallest principal curvature constant in regions where it is of class $C^2$. We prove this result in the Euclidean, spherical and hyperbolic space.