The Banach-Mazur distance is a well-established notion of functional analysis and convex geometry, which quantifies how two normed spaces (or two convex bodies) of the same dimension are close to being isometric (or linearly equivalent). Despite its theoretical importance, many of its non-asymptotic properties remain quite elusive, even in small dimensions. Some of these challenges seem to be caused by the fact that the Banach-Mazur can be surprisingly difficult to compute it, even for concrete and familiar pairs of normed spaces (or convex bodies). During the talk, we will discuss some recent results the Banach-Mazur distance in small dimensions, where the distance is determined precisely. These results include:

• Determination of the Banach-Mazur distance between real $\ell_1$ and $\ell_\infty$ spaces of dimension three and four.
• Construction of the planar convex body, which is different from a triangle and equidistant to every symmetric planar convex body.
• A theorem stating that the Banach-Mazur distance of two three-dimensional cones with arbitrary centrally symmetric bases is equal to the distance of the bases. This leads to an explicit isometric embedding of two-dimensional symmetric Banach-Mazur compactum into the three-dimensional non-symmetric Banach-Mazur compactum.

This is a joint work with Marin Varivoda from University of Zagreb and with Florian Grundbacher from Technical University in Munich.
Meeting Id: 942 8188 0494 Password: 268545