Aequationes mathematicae

, Volume 83, Issue 3, pp 223–237 | Cite as

Numerical verification of condition for approximately midconvex functions

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Abstract

Let X be a normed space and V be a convex subset of X. Let \({\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}\). A function \({f \colon V \to \mathbb{R}}\) is called α-midconvex if
$$f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.$$
It can be shown that every continuous α-midconvex function satisfies the following estimation:
$$f(tx + (1 - t)y) - tf(x)-(1 - t)f(y) \leq \sum_{k=0}^{\infty}\frac{1}{2^k}\alpha(d(2^{kt}\|x - y\|)) \quad {\rm for} \, t \in [0, 1]$$
where \({d(t) := 2{\rm dist}(t, \mathbb{Z})}\) for \({t \in [0, 1]}\). 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) = r p for \({p \in (0, 1)}\), 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.

Mathematics Subject Classification (2010)

Primary 26A51 Secondary 39B62 65G40 

Keywords

Midconvex function convexity numerical verification 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institute of Computer ScienceJagiellonian UniversityKrakówPoland

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