Piecewise Constant Roughly Convex Functions

H.X. Phu, N.N. Hai and P.T. An
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H.X. Phu: Institute of Mathematics
N.N. Hai: University of Hue
P.T. An: Institute of Mathematics

Journal of Optimization Theory and Applications, 2003, vol. 117, issue 2, No 9, 415-438

Abstract: Abstract This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: ρ-convexity (in the sense of Klötzler and Hartwig), δ-convexity and midpoint δ-convexity (in the sense of Hu, Klee, and Larman), γ-convexity and midpoint γ-convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions $$f:\mathbb{R} \to \mathbb{R}$$ satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.

Keywords: Generalized convexity; rough convexity; ρ-convexity; δ-convexity; midpoint δ-convexity; γ-convexity; midpoint γ-convexity; piecewise constant function (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1023/A:1023692025631

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