Journal of Convex Analysis 20 (2013), No. 2, 531--543
Copyright Heldermann Verlag 2013
Strongly Midquasiconvex Functions
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 Zoldak
Institute of Mathematics, University of Rzeszów, Rejtana 16A, 35-959 Rzeszów, Poland
marek_z2@op.pl
[Abstract-pdf]
\def\dim{\mathrm{dim\,\,}} \def\R{{\mathbb R}} \def\e{\varepsilon} Let $V$ be a nonempty convex subset of a normed space $X$ and let $\e>0$ and $p>0$ be given. A function $f: V \to \R$ is called {\em $(\e,p)$-strongly midquasiconvex} if $$ f(\frac{x+y}{2}) \leq \max [f(x), f(y)]-\e(\frac{\|x-y\|}{2})^p \text{\ \ for\ \ } x,y \in V. $$ We call $f$ $p$-strongly midquasiconvex if it is $(\e,p)$-strongly midquasiconvex with a certain $\e>0$. We show that if either $p<1$ and $\dim V=1$ or $p<2$ and $\dim V>1$ then there are no $p$-strongly midquasiconvex functions defined on $V$. On the other hand if $X$ is an inner product space with $\dim X \geq 2$, $p \geq 2$, then there exists an $(1,p)$-strongly midquasiconvex function defined on an arbitrary ball in $X$. \medskip Consequently, the case when $p=1$ and $\dim V=1$ is of a special interest. Under this assumptions we characterize lower semicontinuous $1$-strongly midquasiconvex functions.
Keywords: Quasiconvexity, midquasiconvex function, strongly midquasiconvex function.
MSC: 26B25, 39B62
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