Journal of Convex Analysis 33 (2026), No. 3&4, 875--902
Copyright Heldermann Verlag 2026
The Best Constants in the Strong Convexity of |ξ|p and in the Strong Monotonicity of |ξ|p-2ξ
Antonio Gaudiello
Dipartimento di Matematica e Fisica, Università della Campania "L. Vanvitelli", Caserta, Italy
antonio.gaudiello@unicampania.it
Olivier Guibé
Université Rouen Normandie, CNRS, LMRS, UMR 6085, Rouen, France
olivier.guibe@univ-rouen.fr
Francois Murat
Laboratoire J.-L. Lions, Sorbonne Université, Paris, France
francois.murat@sorbonne-universite.fr
[Abstract-pdf]
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.
Keywords: Strong convexity, strong monotonicity, best constants.
MSC: 26B25.
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