Another n-Dimensional Generalization of Pompeiu’s Theorem

Abstract

A theorem of D. Pompeiu states that the distances from the vertices of an equilateral triangle to any point in its plane can serve as the side lengths of a triangle. In this paper, it is proved that the distances from the vertices of a regular tetrahedron to any point in its affine hull can serve as the areas of the faces of a tetrahedron. The analogous statement for higher-dimensional regular simplices is also established, and several questions are posed for further investigation. Dedicated to Dr. Wasfi al-Kafri for giving the second named author, MH, more than 55 years ago, his own copy of H. G. Forder’s inspiring book A School Geometry, a book that MH will treasure forever.