# Minimal and maximal matrix convex sets

## TL;DR

This paper investigates the relationships between minimal and maximal matrix convex sets associated with convex bodies, providing sharp bounds, characterizations, and explicit dilation constructions, with implications for operator theory and convex geometry.

## Contribution

It establishes sharp bounds for matrix convex set inclusions, characterizes when these sets coincide, and introduces explicit dilation constructions with broad applications.

## Key findings

- Sharp constant for $ heta(ar{B}_{p,d})$ for all $p$
- Characterization of convex bodies with equal minimal and maximal sets as simplices
- Every $d$-tuple of operators can be dilated to commuting operators with norm at most $ extsqrt{d}$

## Abstract

To every convex body $K \subseteq \mathbb{R}^d$, one may associate a minimal matrix convex set $\mathcal{W}^{\textrm{min}}(K)$, and a maximal matrix convex set $\mathcal{W}^{\textrm{max}}(K)$, 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 $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$ hold? For a convex body $K$, we aim to find the optimal constant $\theta(K)$ such that $\mathcal{W}^{\textrm{max}}(K) \subseteq \theta(K) \cdot \mathcal{W}^{\textrm{min}}(K)$; 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 \[ \theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{\textrm{max}}(K) = \mathcal{W}^{\textrm{min}}(K)$ 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.