The Banach-Mazur distance between the cube and the cross-polytope in dimensions three and fourMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

The Banach-Mazur distance is a well-established notion of convex geometry with numerous important applications in the fields like discrete geometry or local theory of Banach spaces. This notion has already been extensively studied by many different authors, but the vast majority of established results are of the asymptotic nature. The non-asymptotic properties of Banach-Mazur distance seem to be quite elusive and even in very small dimensions they are surprisingly difficult to establish. Actually there are rather few situations in which the Banach-Mazur distance between a pair of convex bodies was determined precisely. One example illustrating this difficulty is the case of the cube and the cross-polytope, as their Banach-Mazur distance was known only in the planar case. In this talk we prove that the distance between the cube and the cross-polytope is equal to 9/5 in the dimension three and it is equal to 2 in dimension four.