Quasiconvexity and Density Topology

Rabier, Patrick J.

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 178-187

Résumé

We prove that if $f\,:\,{{\mathbb{R}}^{N}}\,\to \,\overline{\mathbb{R}}$ is quasiconvex and $U\,\subset \,{{\mathbb{R}}^{N}}$ is open in the density topology, then $\underset{U}{\mathop{\sup }}\,f=\text{ess}\,\underset{U}{\mathop{\sup }}\,f$ , while ${{\inf }_{U}}\,f\,=\,\text{ess}\,{{\inf }_{U}}\,f$ if and only if the equality holds when $U\,\subset \,{{\mathbb{R}}^{N}}$ . The first (second) property is typical of $\text{lsc}\,\text{(usc)}$ 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 situations. 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.

Détail

DOI : 10.4153/CMB-2012-028-5

Mots-clés : 52A41, 26B05, density topology, quasiconvex function, approximate continuity, point of continuity

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