On the set of local extrema of a subanalytic function

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 $\mathrm{Max}(f)$ be the set of local maxima of $f$ and consider ${\mathrm{Max}}_{\lambda }(f):=\mathrm{Max}(f)\cap \{f=\lambda \}$ for each $\lambda \in \mathbb{R}$. If $f$ is continuous, then $\mathrm{Max}(f)={⨆}_{\lambda \in \mathbb{R}}{\mathrm{Max}}_{\lambda }(f)\in \mathfrak{F}(M)$ if and only if the family $\{{\mathrm{Max}}_{\lambda }(f){\}}_{\lambda \in \mathbb{R}}$ is locally finite in $M$. If we erase continuity condition, there exist subanalytic functions $f:X\to M$ such that $\mathrm{Max}(f)\in \mathfrak{F}(M)$, but the family $\{{\mathrm{Max}}_{\lambda }(f){\}}_{\lambda \in \mathbb{R}}$ is not locally finite in $M$ or such that $\mathrm{Max}(f)$ 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 $\mathrm{Max}(f)\in \mathfrak{F}(M)$ and the family $\{{\mathrm{Max}}_{\lambda }(f){\}}_{\lambda \in \mathbb{R}}$ 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 $\mathrm{Max}(f)$ does not belong to $\mathfrak{F}(M)$.