Quasiconvexity and Density Topology

Canadian mathematical bulletin

Patrick J. Rabier¹ 

[1] Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

关键词: density topology;    quasiconvex function;    approximate continuity;    point of continuity;   

DOI  :  10.4153/CMB-2012-028-5

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We prove that if $f:mathbb{R}^{N}ightarrow overline{mathbb{R}}$ isquasiconvex and $Usubset mathbb{R}^{N}$ 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 ordinaryopen subset, there seems to be no record that they both hold for allquasiconvex functions.This property ensures that the pointwise extrema of $f$ on any nonemptydensity open subset can be arbitrarily closely approximated by values of $f$achieved on ``large'' subsets, which may be of relevance in a variety ofissues. To support this claim, we use it to characterize the common pointsof continuity, or approximate continuity, of two quasiconvex functions thatcoincide away from a set of measure zero.

【 授权许可】

Unknown