A Helly-type theorem for semi-monotone sets and monotone maps (Q393726)

In [\textit{S. Basu} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 107, No. 1, 5--33 (2013; Zbl 1273.14123); J. Eur. Math. Soc. (JEMS) 15, No. 2, 635--657 (2013; Zbl 1284.14081)], the authors of the present paper introduced a class of definable subsets of \(\mathbb R^n\), called \textit{semi-monotone sets}, and definable maps, called \textit{monotone maps}, in an o-minimal structure over \(\mathbb R\). Real semi-algebraic sets are examples of semi-monotone sets. Semi-monotone sets, and more generally graphs of monotone maps, are in general non-convex, but they behave in many ways like convex subsets of \(\mathbb R^n\). In the paper under review, the authors prove a version of the classical Helly's theorem [\textit{E. Helly}, Monatsh. Math. Phys. 37, 281--302 (1930; JFM 56.0499.03)] for graphs of monotone maps: Let \(\mathcal F=\{ F_i\}_{i\in I}\) be a family of definable subsets of \(\mathbb R^n\); for any \(J\subseteq I\), let \(\mathcal F_J\) denote the set \(\bigcap_{j\in J} F_j\) and \(|J|\) denote the cardinality of \(J\). Suppose that, for each \(i\in I\), the set \(F_i\) is the graph of a monotone map, and for each \(J\subset I\), with \(|J|\leq n+1\), \(\mathcal F_J\) is non-empty and the graph of a monotone map. Then \(\mathcal F_I\) is non-empty and the graph of a monotone map as well. Moreover, if \(\dim \mathcal F_J\geq d\) for each \(J\subset I\), with \(|J|\leq n+1\), then \(\dim \mathcal F_I\geq d\).