  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.
