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- W2123689469 abstract "We generalize the Hadwiger theorem on transversals to collections of compact convex sets in the plane to the case where the sets are connected and the transversals form an arrangement of pseudolines. The proof uses the embeddability of pseudoline arrangements in topological affine planes. In 1940 Santalo showed (12), by an example, that Vincensini's proof (13) of an extension of Helly's theorem was incorrect. Vincensini c laimed to have proven that for any finite collection S of at least three compact convex sets in the plane, any three of which were met by a line, there must exist a meeting all the sets. This would have constituted an extension of the planar Helly theorem (10), which showed that the same assertion holds if line is replaced by point. The Santalo example was later extended by Hadwiger and Debrunner (9) to show that even if the convex sets are disjoint the conclusion still may not hol d. In 1957, however, Hadwiger showed that the conclusion of the theorem is valid if the hypothesis is strengthened by imposing a consistency condition on the order in which the triples of sets are met by transversals:" @default.
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- W2123689469 date "2007-01-01" @default.
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- W2123689469 title "The Hadwiger Transversal Theorem for Pseudolines" @default.
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