How Does the Contraction Property Fail for Convex Functions on Normed Spaces?

On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT(0)-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces.