  We study contractivity properties of gradient flows for functions on normed
spaces or, more generally, on Finsler manifolds. Contractivity of the flows
turns out to be equivalent to a new notion of convexity for the functions. This
is different from the usual convexity along geodesics in non-Riemannian Finsler
manifolds. As an application, we show that the heat flow on Minkowski normed
spaces other than inner product spaces is not contractive with respect to the
quadratic Wasserstein distance.
