  A polytope in a finite-dimensional normed space is subequilateral if the
length in the norm of each of its edges equals its diameter. Subequilateral
polytopes occur in the study of two unrelated subjects: surface energy
minimizing cones and edge-antipodal polytopes. We show that the number of
vertices of a subequilateral polytope in any d-dimensional normed space is
bounded above by (d/2+1)^d for any d >= 2. The same upper bound then follows
for the number of vertices of the edge-antipodal polytopes introduced by
I.Talata (Period. Math. Hungar. 38 (1999), 231--246). This is a constructive
improvement to the result of A.P\'or (to appear) that for each dimension d
there exists an upper bound f(d) for the number of vertices of an
edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space
the only subequilateral polytopes are equilateral simplices.
