Diffeomorphism groups of compact convex sets

For K⊆Rn a compact convex subset with non-empty interior, let Diff∂K(K) be the group of all C∞-diffeomorphisms of K which fix ∂K pointwise. We show that Diff∂K(K) is a C0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.