Quasiconvexity and density topology

TL;DR

This paper demonstrates that quasiconvex functions exhibit properties similar to lower and upper semicontinuous functions within the density topology, with implications for understanding their extrema and points of continuity.

Contribution

It establishes that quasiconvex functions' supremum and infimum over density open sets equal their essential bounds, a property previously unrecorded for all such functions.

Findings

Sup_U f = ess sup_U f for quasiconvex functions in density topology

Inf_U f = ess inf_U f iff it holds for U = R^N

Characterization of points of continuity for quasiconvex functions

Abstract

We prove that if f : R^N --> R is quasiconvex and U is open in the density topology of R^N, then sup_U f = ess sup_U f, while inf_U f = ess inf_U f if and only if the equality holds when U = 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.