ON THE MAZUR-ULAM THEOREM AND THE SOLUTION OF TWO PROBLEMS OF RASSIAS

Abstract

In this paper, we show that a mapping which preserves the area of any planar convex hull of any points in [0, 1]3, or preserves the volume of any convex hull of any points in [0, 1]3, must be an isometric affine, which is a composition of some special isometries. This provides a solution to two problems that were posed by Rassias.