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A Helly-type theorem for semi-monotone sets and monotone maps

<13>Semi-monotone sets and graphs of monotone maps are topologically regular cells generalizing (and having some properties of) bounded convex sets. Unlike convex sets, the intersection of two graphs may not be a regular cell, it may even have arbitrarily large number of connected components. The main theorem of this paper is a far-reaching generalization of the classical Helly theorem on intersection of convex sets in a direction not explored earlier. This generalization is important in computational geometry, where rotation of the axes is sparsity-destroying and very expensive.