Abstract. Let $P$ be a convex $d$-polytope in $E^d$ such that any point $x \in P$ belongs to finitely many affine diameters of $P$. Then the set of points from $P$, each belonging to an even number of affine diameters of $P$, has $d$-dimensional measure zero.

AMS Subject Classification (1991): 52A20, 52B11

Keyword(s): Antipodal faces, affine diameter, convex polytope

Received September 11, 2002, and in revised form March 6, 2003. (Registered under 2906/2009.)