The product of the distances of a point inside a regular polytope to its vertices

We show that the maximum of the product of the distances from a point inside an n-dimensional regular simplex, cross-polytope or cube to the vertices is attained at the midpoint of an edge for small n, but is attained at symmetrically placed pairs on an edge for sufficiently high dimensions. We also examine the problem for regular polygons and general triangles in the plane. Murray Klamkin passed away on August 15, 2004. He was Professor of Mathematics at the University of Alberta, Edmonton, Alberta, Canada.