# banach spaces – Uniform smoothness and twice-differentiability of norms

To get to the simplest case, consider a norm $$|cdot|$$ over $$R^n$$ that is uniformly convex of power-type 2, that is, there is a constant $$C$$ such that $$frac{|x+y| + |x – y|}{2} le 1 + C |y|^2$$ for all $$x$$ with $$|x| = 1$$ and for all $$y$$.

Question: Does this guarantee that $$|cdot|$$ has a second-order Taylor expansion on $$R^n setminus {0}$$, that is, there is a vector $$g$$ and a symmetric matrix $$A$$ such that $$|x + y| = |x| + langle g, y rangle + frac{1}{2} langle Ay, y rangle + o(|y|^2)$$ for all $$x neq 0$$. (Apparently this is a weaker requirement than twice-differentiability of $$|cdot|$$ on $$R^n setminus {0}$$)

It is easy to see that $$|cdot|$$ is differentiable on $$R^n setminus {0}$$, and a classic result of Alexandrov guarantees that the above second-order Taylor expansion holds for any convex function on almost every point $$x$$. It is also known that the norm of any separable Banach space can be approximated arbitrarily well by a power-type 2 norm that is twice differentiable on $$R^n setminus {0}$$ (see Lemma 2.6 here). But I wonder if the original norm itself has a second-order Taylor expansion.