  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 Carath\'{e}odory's and of Minkowski's theorem
for matrix convex sets.
