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Quasiconvexity and Density Topology
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 178-187
- Print publication:
- 14 March 2014
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- Article
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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.
