1. Introduction Let X be a set of points in Euclidean d-space Ed. Then conv X and aff X denote, respectively, the convex hull and the affine hull of X. Two points x and y are called antipodal points of X if there are distinct parallel supporting hyperplanes of conv X, one of which contains x and the other contains y. We say that X is an antipodal set if any two points of X are antipodal points of X. In the case that X is a convex d-polytope P, a related notion was recently introduced in [Talata 1999]. P is an edge-antipodal d-polytope if any two vertices of P, that lie on an edge of P, are antipodal points of P.

According to a well-known result of Danzer and Grünbaum [1962], conjectured independently by Erdős [1957] and Klee [1960], the cardinality of any antipodal set in Ed is at most 2d. Talata [1999] conjectured that there exists a smallest positive integer m such that the cardinality of the vertex set of any edge-antipodal 3-polytope is at most m. In an elegant paper, Csikos [2003] showed that m ≤ 12. In this paper, we prove that m = 8. THEOREM. The number of vertices of an edge-antipodal 3-polytope P is at most eight, with equality only if P is an affine cube.