A coordinate cone in is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone set is a topological regular cell.
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Saugata Basu, Andrei Gabrielov, Nicolai Vorobjov, Semi-monotone sets. J. Eur. Math. Soc. 15 (2013), no. 2, pp. 635–657DOI 10.4171/JEMS/369