On the set of local extrema of a subanalytic function

TL;DR

This paper investigates the structure and properties of local extrema of subanalytic functions within certain categories of subanalytic sets, highlighting conditions under which the set of maxima is subanalytic or has other topological features.

Contribution

It characterizes when the set of local maxima of subanalytic functions belongs to a given subanalytic category, especially analyzing the effects of continuity and the structure of the function.

Findings

The set of local maxima is subanalytic if the family of level maxima is locally finite.

Continuity of the function influences the subanalyticity of the maxima set.

Counterexamples show that without continuity, maxima sets may not be subanalytic or locally finite.

Abstract

Let ${\mathfrak F}$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical and basic topological operations. Let $M$ be a real analytic manifold and denote ${\mathfrak F}(M)$ the family of the subsets of $M$ that belong to ${\mathfrak F}$. Let $f:X\to{\mathbb R}$ be a subanalytic function on a subset $X\in{\mathfrak F}(M)$ such that the inverse image under $f$ of each interval of ${\mathbb R}$ belongs to ${\mathfrak F}(M)$. Let ${\rm Max}(f)$ be the set of local maxima of $f$ and consider ${\rm Max}_\lambda(f):={\rm Max}(f)\cap\{f=\lambda\}$ for each $\lambda\in{\mathbb R}$. If $f$ is continuous, then ${\rm Max}(f)=\bigsqcup_{\lambda\in{\mathbb R}}{\rm Max}_\lambda(f)\in{\mathfrak F}(M)$ if and only if the family $\{{\rm Max}_\lambda(f)\}_{\lambda\in{\mathbb R}}$ is locally finite in $M$. If we erase continuity condition, there exist subanalytic functions $f:X\to M$ such that ${\rm Max}(f)\in{\mathfrak F}(M)$, but the family $\{{\rm Max}_\lambda(f)\}_{\lambda\in{\mathbb R}}$ is not locally finite in $M$ or such that ${\rm Max}(f)$ is connected but it is not even subanalytic. If ${\mathfrak F}$ is the category of subanalytic sets and $f:X\to{\mathbb R}$ is a subanalytic map $f$ that maps relatively compact subsets of $M$ contained in $X$ to bounded subsets of ${\mathbb R}$, then ${\rm Max}(f)\in{\mathfrak F}(M)$ and the family $\{{\rm Max}_\lambda(f)\}_{\lambda\in{\mathbb R}}$ is locally finite in $M$. If the category ${\mathfrak F}$ contains the intersections of algebraic sets with real analytic submanifolds and $X\in{\mathfrak F}(M)$ is not closed in $M$, there exists a continuous subanalytic function $f:X\to{\mathbb R}$ with graph belonging to ${\mathfrak F}(M\times{\mathbb R})$ such that inverse images under $f$ of the intervals of ${\mathbb R}$ belong to ${\mathfrak F}(M)$ but ${\rm Max}(f)$ does not belong to ${\mathfrak F}(M)$.