Approximately Midconvex Functions

Krzysztof Misztal (), Jacek Tabor () and Józef Tabor ()
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Krzysztof Misztal: Jagiellonian University
Jacek Tabor: Jagiellonian University
Józef Tabor: University of Rzeszów

Chapter Chapter 14 in Functional Equations in Mathematical Analysis, 2011, pp 177-190 from Springer

Abstract: Abstract In the paper we propose very general definition of approximate midconvexity. Let α: [0, ∞) → ℝ be a given function. Let X be a normed space and V a convex subset of X. A function f: V → ℝ will be called α(⋅) - midconvex if $$f\left (\frac{x + y} {2} \right ) \leq \frac{1} {2}f(x) + \frac{1} {2}f(y) + \alpha (\|x - y\|)\mbox{ for }x,y \in V.$$ The above definition simultaneously generalizes approximate and uniform midconvexities. We present several results concerning this notion.

Keywords: Approximately convex function; Approximately midconvex function; Semiconvex function (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-1-4614-0055-4_14

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