Frank Morgan

On March 29 at Banff, Mohammad Ghomi talked on his proof [CGR] with Choe and Ritoré of the isoperimetric inequality in the complement of a convex body K in Rⁿ: the area of a hypersurface enclosing volume V outside the convex body is at least the area of a hemisphere of volume V. I asked whether it suffices to assume K mean convex (nonnegative mean curvature). The answer is no. The first counterexample, provided by Cantarella and Ghomi, was K a catenoid and the hypersurface a portion of a cylinder of large radius. What about for small topological balls? The answer is still no. Otherwise a small ball split in half by a minimal surface would have to have equal perimeter on each side, each half would be isoperimetric, the sphere would meet the minimal surface normally, and the minimal surface would have to be a cone.

[CGR] Choe, Jaigyoung(KR-KIAS-SM); Ghomi, Mohammad(1-GAIT); Ritoré, Manuel(E-GRANS-GT)
The relative isoperimetric inequality outside convex domains in Rⁿ. Calc. Var. Partial Differential Equations 29 (2007), no. 4, 421-429.

P.S. May, 2010. Pacific Journal (Vol. 244, No. 2, 2010) has an article by Mouhamed Moustapha Fall on “Area-minimizing regions with small volume in Riemannian manifolds with boundary.” The main result is that inside a smooth, bounded region with boundary in Rⁿ or in any Riemannian manifold, small isoperimetric regions are approximate half-balls near boundary points of maximum mean curvature H. (Should also apply to unbounded region such as complement of smooth body.) Moreover, if another domain has H’ < H, then for small volume the isoperimetric profiles satisfy I’ > I. In particular, outside a smooth strictly mean convex 3D region, for small volumes, area is greater than that of the Euclidean half-ball of the same volume. This does not contradict the nearly opposite conclusion we came to above because of the strict inequality in the hypothesis H’ < H and the restriction to small volumes.