ON THE BANACH-MAZUR DISTANCE BETWEEN THE CUBE AND THE CROSSPOLYTOPE

摘要

In this note we study the Banach-Mazur distance between the n -dimensional cube and the crosspolytope. Previous work shows that the distance has order root n, and here we will prove some explicit bounds improving on former results. Even in dimension 3 the exact distance is not known. and based on computational results it is conjectured to be 9/5. Here we will also present computer based potentially optimal results in dimension 4 to 8.