Let M be a convex body, i.e., a compact, convex set with non-empty interior, in n-dimensional Euclidean space En. A chord [a, b] of M is said to be an affine diameter of M, if, and only if, there exists a pair of (different) parallel supporting hyperplanes of that body, each containing one of the points a, b. The following result of Eggleston (cf. [1] and [2]) is well-known. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to exactly three affine diameters. In [3] this result is sharpened. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to at least two, but a finite number of affine diameters. A natural problem for the n-dimensional case, based on Eggleston's result, is the following (cf. also [4]). Is it true that the n-dimensional simplex is the only convex body in En such that through each interior point pass precisely 2n − l affine diameters? For the case of convex polytopes, i.e., convex bodies with a finite number of extreme points, we shall give a positive answer to this question.