Let $d$ be a positive integer. Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives. Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is denoted by $H:\mathbb{R}^d\to\mathbb{R}$ and defined by: $$ H(p)=\sup_{\alpha\in\mathbb{R}^d} p\cdot\alpha - L(\alpha).$$ Assume that there exists a unique $\alpha\in\mathbb{R}^d$ achieving the maximum in the definition of $H(p)$ for any $p\in\mathbb{R}^d$.
Is it possible to conclude that $L$ is strictly convex? We did not assume that $L$ is convex.
A quick drawing makes me think that the latter conclusion holds in dimension $d=1$.
Any help would be appreciated. Please let me know if you think about a reference which may contain the answer.
Best.
