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Quasiconvexity and Density Topology

  Published:2012-09-21
 Printed: Mar 2014
  • Patrick J. Rabier,
    Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
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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.
Keywords: density topology, quasiconvex function, approximate continuity, point of continuity density topology, quasiconvex function, approximate continuity, point of continuity
MSC Classifications: 52A41, 26B05 show english descriptions Convex functions and convex programs [See also 26B25, 90C25]
Continuity and differentiation questions
52A41 - Convex functions and convex programs [See also 26B25, 90C25]
26B05 - Continuity and differentiation questions
 

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