Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}$$\end{document}. A function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \colon V \to \mathbb{R}}$$\end{document} is called α-midconvex if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.$$\end{document}It can be shown that every continuous α-midconvex function satisfies the following estimation:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(tx + (1 - t)y) - tf(x)-(1 - t)f(y) \leq \sum_{k=0}^{\infty}\frac{1}{2^k}\alpha(d(2^{kt}\|x - y\|)) \quad {\rm for} \, t \in [0, 1]$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d(t) := 2{\rm dist}(t, \mathbb{Z})}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t \in [0, 1]}$$\end{document}. It is an important problem to verify for which functions α the above estimation is optimal. The conjecture of Páles that this is the case for functions of type α(r) = rp for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p \in (0, 1)}$$\end{document}, was proved by Makó and Páles (J Math Anal Appl 369:545–554, 2010). In this paper we present a computer assisted method to verify the optimality of this estimation in the class of piecewise linear functions α.