Let $H$ be a real Hilbert space with norm $\|\cdot\|_H$ (i.e. $\|\cdot\|_H$ is generated by the scalar product native to $H$).
Does there exist another (not necessarily Hilbertian) norm $\lvert\lvert\lvert\cdot\rvert\rvert\rvert$ on $H$ such that $\lvert\vert\lvert\cdot\rvert\rvert\rvert^2$ is not strongly convex?
As a reminder, $f\colon H\to\mathbb{R}$ is strongly convex if there exists $\beta\in\mathbb{R}_{++}$ such that $f-\frac{\beta}{2}\|\cdot\|_H^2$ is convex.
My suspicion is that a counterexample exists even in $\mathbb{R}^N$ (under the Euclidean norm) or $\mathbb{R}^{N\times N}$ (under the Frobenius norm), but I'm having some trouble brainstorming
Best Answer
There is a counterexample on $\mathbb{R}^2$ with $f=\|\cdot\|_1^2$. This graph demonstrates that $f-\frac{\beta}{2}\|\cdot\|_H^2$ is nonconvex.