From:propp@math.mit.edu (Jim Propp)Newsgroups:sci.math.researchSubject:antipodesDate:24 Jan 96 03:10:35 GMTOrganization:MIT Department of Mathematics

Given a centrally symmetric convex compact body K in 3-space and a point p on its surface, the point on the surface farthest from p in the surface metric need not be the antipode of p. (Cf. the Knuth-Kotani puzzle, which asks one to find the point on the surface of a 1-by-1-by-2 box that is farthest from one of the corners; it is not the opposite corner.) However, one might still conjecture that the maximum distance between two points in the surface metric, as BOTH of them vary over the surface, is achieved by a pair of antipodal points. (This is believed to be true in the case of an a-by-b-by-c box for all a, b, and c, though to my knowledge no proof is known even for this very special case.) Can anyone prove or disprove the conjecture? (Is the convexity condition necessary?) Jim Propp Department of Mathematics M.I.T.

Date:Sat, 2 Feb 2002 10:45:37 -0800 (PST)From:Costin Vilcu <costin_v@yahoo.com>Subject:antipodes on convex surfacesTo:eppstein@ics.uci.edu

Dear Professor Eppstein, I would like to inform you that the open problem of Jim Propp you propose at http://www.ics.uci.edu/eppstein/junkyard/open.html concerning antipodes on convex surfaces has been solved. By Proposition 6 (pp. 273) in Geometriae Dedicata 79: 267-275 (2000), two points realizing the intrinsic diameter of a centrally symmetric convex surface must be symmetric (under the same symmetry). The proof holds more generally, for any centrally symmetric surface of genus 0. I proposed there the following problem: Prove or disprove that, if on a centrally symmetric convex surface every point x has an unique farthest point F(x) then x and F(x) must be symmetric (under the same symmetry). (Notice that on boxes there are points with two farthest points.) Meanwhile this problem was also solved -getting a negative answer- and appears in a short joint paper written with Prof. T. Zamfirescu, my thesis advisor. This paper is not yet published. Sincerely Yours, Costin VILCU