# Convex conjugate: Lipschitz continuity of the argmax function

Let $$s,\delta\in\mathbb{R}^{N}$$, $$S\subseteq\mathbb{R}^{N}$$ be a compact convex set, $$f:S\rightarrow\mathbb{R}$$ be a twice differentiable strictly convex function on $$S$$ and $$s\left(\delta\right)=\arg\max_{s\in S}s\cdot\delta-f\left(s\right).$$ Notice that this is related to the definition of convex conjugate of $$f$$: $$f^*\left(\delta\right)\equiv\max_{s\in S}s\cdot\delta-f\left(s\right).$$ Being $$f$$ strictly convex, $$s\left(\cdot\right)$$ is single valued, and it is continuous by the maximum theorem. My question: is $$s\left(\cdot\right)$$ Lipschitz continuous? If not, which additional conditions do I need to get Lipschitz continuity?

The (global) Lipschitz continuity should be equivalent to $$f$$ being strongly convex. In order to find a counterexample, you should try some $$f$$ which is not strongly convex.