I think Cauchy proved the 2-dimensional version of this by averaging orthogonal projections over all different directions. So there may be a proof like that in higher dimensions. Convex surfaces are almost everywhere C^2, which may help. —Dan Jim Propp wrote ----- I believe the two-dimensional version of this assertion was made by Archimedes, and was intended to serve as a way of characterizing arc-length. On Sun, Jun 23, 2019 at 8:09 PM Adam P. Goucher <apgoucher@gmx.com> wrote: ----- Excellent proof! By the way, for the 'obvious' lemma that the girth is an increasing function of the two semi-axes, it follows from: Theorem: If L is a compact subset of R^n and K is a convex compact subset of L, then the surface area of K is bounded above by the surface area of L, with equality if and only if K = L. ... ----- -----