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 operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote ${\mathfrak {F}}(M)$ the family of the subsets of M that belong to the category ${\mathfrak {F}}$. Let $f:X\rightarrow \mathbb {R}$ 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 its level sets $\mathrm{Max}_\lambda (f):=\mathrm{Max}(f)\cap \{f=\lambda \}=\{f=\lambda \}{\setminus }{\text {Cl}}(\{f>\lambda \})$ for each $\lambda \in \mathbb {R}$. In this work we show that if f is continuous, then $\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$ 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 $f:X\rightarrow M$ 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. We show in addition that if ${\mathfrak {F}}$ is the category of subanalytic sets and $f:X\rightarrow \mathbb {R}$ is a (non-necessarily continuous) 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. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if ${\mathfrak {F}}$ is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an ${\mathfrak {F}}$-subset of M of an analytic function on M, then the family $\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$ is locally finite in M and $\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$. We also show that 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, then there exists a continuous subanalytic function $f:X\rightarrow \mathbb {R}$ 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 ${\mathfrak {F}}(M)$. As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function $f:X\rightarrow \mathbb {R}$ coincides with the set of local extrema $\mathrm{Extr}(f):=\mathrm{Max}(f)\cup \mathrm{Min}(f)$. This means that if $f:X\rightarrow \mathbb {R}$ is a continuous subanalytic function defined on a closed set $X\in {\mathfrak {F}}(M)$ such that the inverse image under f of each interval of $\mathbb {R}$ belongs to ${\mathfrak {F}}(M)$, then the set $\mathrm{Op}(f)$ of openness points of f belongs to ${\mathfrak {F}}(M)$. Again the closedness of X in M is crucial to guarantee that $\mathrm{Op}(f)$ belongs to ${\mathfrak {F}}(M)$. The type of results stated above are straightforward if ${\mathfrak {F}}$ is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category ${\mathfrak {F}}$ of subanalytic sets that does not constitute an o-minimal structure.