```From:           propp@math.mit.edu (Jim Propp)
Newsgroups:     sci.math.research
Subject:        antipodes
Date:           24 Jan 96 03:10:35 GMT
Organization:   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 surfaces
To:             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
```