Minimal and maximal matrix convex sets

Abstract

To every convex body $K\subseteq {\mathbb{R}}^d$, one may associate a minimal matrix convex set ${\mathcal{W}}^{\text{min}}(K)$, and a maximal matrix convex set ${\mathcal{W}}^{\text{max}}(K)$, which have $K$ as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies $K,L\subseteq {\mathbb{R}}^d$ does ${\mathcal{W}}^{\text{max}}(K)\subseteq {\mathcal{W}}^{\text{min}}(L)$ hold? For a convex body $K$, we aim to find the optimal constant $\theta (K)$ such that ${\mathcal{W}}^{\text{max}}(K)\subseteq \theta (K)\cdot {\mathcal{W}}^{\text{min}}(K)$; we achieve this goal for all the ${\ell }^p$ unit balls, as well as for other sets. For example, if ${\overline{\mathbb{B}}}_{p,d}$ is the closed unit ball in ${\mathbb{R}}^d$ with the ${\ell }^p$ norm, then \[ θ(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is sharp, and it is new for all $p\ne 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which ${\mathcal{W}}^{\text{max}}(K)\subseteq {\mathcal{W}}^{\text{min}}(L)$. In particular, we obtain that a convex body $K$ satisfies ${\mathcal{W}}^{\text{max}}(K)={\mathcal{W}}^{\text{min}}(K)$ if and only if $K$ is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every $d$-tuple of self-adjoint operators of norm less than or equal to $1$, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.

47 pages, 5 figures. Version 2 corrects minor errors and clarifies some of the theorem statements. Remarks have been added throughout the text, and section 4 has been expanded a bit. To appear in Journal of Functional Analysis

References in corpus (1)

• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

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