@RobertBryant, the Effimov paper which you cite is only the announcement of the result. The full proof is presented in Mat. Sb. (N.S.) 64(106) (1964), 286–320, which is 35 pages.

@Gerry Myerson: One can think of it that way, subject to a convexity constraint, and maybe computers can be used to verify the conjectured inequality for polytopes with few vertices, which would certainly constitute progress. But ultimately I think it would be more natural to think of the problem in terms of the isoperimetric inequality.