## Abstract

For every convex body K⊆R^{d}, there is a minimal matrix convex set W^{min}(K), and a maximal matrix convex set W^{max}(K), which have K as their ground level. We aim to find the optimal constant θ(K) such that W^{max}(K)⊆θ(K)⋅W^{min}(K). For example, if B‾_{p,d} is the unit ball in R^{d} with the ℓ^{p} norm, then we find that θ(B‾_{p,d})=d^{1−|1/p−1/2|}. This constant is sharp, and it is new for all p≠2. Moreover, for some sets K we find a minimal set L for which W^{max}(K)⊆W^{min}(L). In particular, we obtain that a convex body K satisfies W^{max}(K)=W^{min}(K) only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most d. We also introduce new explicit constructions of these (and other) dilations.

Original language | English |
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Pages (from-to) | 3197-3253 |

Number of pages | 57 |

Journal | Journal of Functional Analysis |

Volume | 274 |

Issue number | 11 |

DOIs | |

State | Published - 1 Jun 2018 |

Externally published | Yes |

## Keywords

- Abstract operator system
- Dilation
- Matrix convex set
- Matrix range