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

Florian Grundbacher, Tomasz Kobos

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.

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

Abstract

We determine certain Banach-Mazur distances involving -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 to Euclidean space in terms of the distances of the spaces to Euclidean space. For we show that if , then . Interpreting -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 . We further show that in dimension 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 . As consequences, we obtain an explicit isometric embedding of the -dimensional symmetric Banach-Mazur compactum into the -dimensional (non-symmetric) compactum and lift a recent construction of arbitrarily large equilateral sets in the -dimensional symmetric compactum to all higher dimensions.
Paper Structure (5 sections, 16 theorems, 124 equations, 4 figures)

This paper contains 5 sections, 16 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction and Results
  2. The Distance of lp-Sums to the Euclidean Space
  3. The Distance of (Double) Cones to l1n and the Simplex
  4. The Distance of (Double) Cones With Arbitrary Planar Symmetric Bases
  5. Acknowledgements

Key Result

Theorem 1.1

Let $n_1, n_2, \ldots, n_k \geq 1$ be integers and let $X_i$ be an $n_i$-dimensional real normed space for $i=1, \ldots, k$. Then, for any $p \in [1,\infty]$, where $r = \frac{2p}{|p-2|} \in [2,\infty]$ (with the conventions $r = \infty$ when $p=2$ and $r=2$ when $p=\infty$).

Figures (4)

  • Figure 1: An example of the situation in Lemma \ref{['lem:triangles']} and its proof for $d = 1.4$: $T$ (red, dashed), $S$ (blue, solid), $d T$ (orange, dotted).
  • Figure 2: An example of the situation in Case $2$ in the proof of Theorem \ref{['thm:3d_cones']}: $C_1$ (red), $C_2$ (blue), $d C_2$ (orange). The vertex of the middle cone is projected onto the projection image of the cone's base.
  • Figure 3: An example of the situation for condition (i) of Lemma \ref{['lem:triangles']} in Case $3$ in the proof of Theorem \ref{['thm:3d_cones']}: $C_1$ (red), $C_2$ (blue), $d C_2$ (orange). The triangles $T = C_1 \cap V$, $S = C_2 \cap V$, and $d T$ are marked as dotted sections of the respective cones. The projection along the first coordinate axis in the chosen coordinate system in $V$ maps the middle cone to a set almost affinely equivalent to its base.
  • Figure 4: An example of the situation for condition (ii) of Lemma \ref{['lem:triangles']} in Case $3$ in the proof of Theorem \ref{['thm:3d_cones']}: $C_1$ (red), $C_2$ (black), $d C_2$ (orange). The projection with kernel parallel to $w$ maps the inner and outer cones to sets almost affinely equivalent to their bases.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 2.1
  • ...and 20 more