Georgia Institute of Technology College of Sciences

We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A. Giannopoulos, "A note to the Banach-Mazur distance to the cube", where he proves that $R_{\infty} ^n < c n^{5/6}$.