Abstract view
Quasiconvexity and Density Topology


Published:20120921
Printed: Mar 2014
Patrick J. Rabier,
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Abstract
We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.