On the set of local extrema of a subanalytic function

• Autores: José Francisco Fernando Galván
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 71, Fasc. 1, 2020, págs. 1-24
• Idioma: inglés
• DOI: 10.1007/s13348-019-00245-6
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• Resumen

  • 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.