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.