Abstract

Let VV be a nonempty convex subset of a normed space XX and let ε>0\e>0 and p>0p>0 be given. A function f:V→Rf: V \to \R is called (ε,p)(\e,p)-strongly midquasiconvex if f(x+y2)≤max⁡[f(x),f(y)]−ε(∥x−y∥2)p  for  x,y∈V.f(\frac{x+y}{2}) \leq \max [f(x), f(y)]-\e(\frac{\|x-y\|}{2})^p \text{\ \ for\ \ } x,y \in V. We call ff pp-strongly midquasiconvex if it is (ε,p)(\e,p)-strongly midquasiconvex with a certain ε>0\e>0. We show that if either p<1p<1 and dim  V=1\dim V=1 or p<2p<2 and dim  V>1\dim V>1 then there are no pp-strongly midquasiconvex functions defined on VV. On the other hand if XX is an inner product space with dim  X≥2\dim X \geq 2, p≥2p \geq 2, then there exists an (1,p)(1,p)-strongly midquasiconvex function defined on an arbitrary ball in XX. Consequently, the case when p=1p=1 and dim  V=1\dim V=1 is of a special interest. Under this assumptions we characterize lower semicontinuous 11-strongly midquasiconvex functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Jacek Tabor

Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Poland

tabor@ii.uj.edu.pl

Józef Tabor

Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland

tabor@univ.rzeszow.pl

Marek Żołdak

Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland

marek_z2@op.pl

Keywords

  • Quasiconvexity
  • midquasiconvex function
  • strongly midquasiconvex function

Mathematics Subject Classification

26B25, 39B62

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Suggested citation

J. Tabor, J. Tabor, M. Żołdak. “Strongly Midquasiconvex Functions.” Journal of Convex Analysis 20 (2013), No. 2, 531–543. https://doi.org/10.68381/jca20032

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.