LAUSR

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.