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Separoids, their categories and a Hadwiger-type theorem for transversals. (English) Zbl 1002.52008

If any 3 sets in a family of convex sets in \(\mathbb R^2\) have a common transversal (line) then the family does not necessarily have a common transversal. According to H. Hadwiger [Port. Math. 16, No. 2, 23-29 (1957; Zbl 0081.16404)] it has, if the family satisfies a suitable order property. For an extension to \(\mathbb R^n\) see, amongst others, J. Goodman and R. Pollack [J. Am. Math. Soc. 1, No. 2, 301-309 (1988; Zbl 0642.52003)]. The authors propose a different such condition and prove a far-reaching (Borsuk-Ulam-type) generalization of Hadwiger’s theorem. (What makes this article interesting is the connection between convex geometry and topology).

MSC:

52A35 Helly-type theorems and geometric transversal theory
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