It is shown that a convex d-polytope P (d\(\geq 2)\) is a simplex if and only if each interior point of P belongs to precisely \(2^ d-1\) affine diameters of P. (An affine diameter of P is a chord whose endpoints lie in different parallel supporting hyperplanes of that polytope). This generalizes, at least for convex bodies with a finite number of extreme points, a planar results of H. G. Eggleston [Problems in Euclidean space. Applications of convexity. (1957; Zbl 0083.381)].