Exact Banach-Mazur distances of certain $\ell_p$-sums and cones

Abstract

We determine certain Banach-Mazur distances involving $\ell_p$-direct sums of finite-dimensional real normed spaces and related cone constructions of convex bodies. Using a recent characterization of the optimal Banach-Mazur position with respect to the Euclidean ball, we derive a closed formula for the distance from $X_1 \oplus_p \cdots \oplus_p X_k$ to Euclidean space in terms of the distances of the spaces $X_i$ to Euclidean space. For $p = 1$ we show that if $d_{BM}(X,\ell_1^n) \leq 3$, then $d_{BM}(X \oplus_1 \ell_1^m, \ell_1^{n+m}) = d_{BM}(X,\ell_1^n)$. Interpreting $\ell_1$-sums geometrically as double cones motivates a study of single cones over arbitrary convex bases, for which we establish an analogous result with the simplex replacing $\ell_1$. We further show that in dimension $3$ the distance between single cones with symmetric bases equals the distance between the bases, and that the same equality holds for double cones over planar symmetric bases in arbitrary dimension, under an additional assumption on the distance of the bases to $\ell_1^2$. As consequences, we obtain an explicit isometric embedding of the $2$-dimensional symmetric Banach-Mazur compactum into the $3$-dimensional (non-symmetric) compactum and lift a recent construction of arbitrarily large equilateral sets in the $2$-dimensional symmetric compactum to all higher dimensions.