Dilation theory in finite dimensions and matrix convexity

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most 1, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Caratheodory's and of Minkowski's theorems for matrix convex sets.