The Best Constants in the Strong Convexity of |ξ|^(p) and in the Strong Monotonicity of |ξ|^(p-2)ξ

We prove that the function $\xi\in \mathbb{R}^N\rightarrow \vert \xi\vert^p$ is strongly convex for every $p$, $p\in]1,+\infty[$, and we give a characterization, through the resolution of an algebraic equation, of the best constant which appears in the strong convexity inequality. The same proof allows us to prove that the function $\xi\in \mathbb{R}^N\to \vert \xi\vert^{p-2}\xi$ is strongly monotone and to identify the explicit value of the best constant.