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 general principle to deduce theorems from their classical infinite-dimensional counterparts. In addition providing unified proofs known theorems, we versions Agler's rational an annulus, Berger's operators numerical radius at most $1$, and Putinar-Sandberg range theorem. As key tool, prove Carath\'{e}odory's Minkowski's matrix convex sets.