The Hadwiger Transversal Theorem for Pseudolines

Summary

We generalize the Hadwiger theorem on line 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.

Santalo [1940] showed, by an example, that Vincensini's proof [1935] of an extension of Helly's theorem was incorrect. Vincensini claimed to have proved that for any finite collection S of at least three compact convex sets in the plane, any three of which are met by a line, there must exist a line meeting all the sets. This would have constituted an extension of the planar Helly theorem [Helly 1923] to the effect that the same assertion holds if “line” is replaced by “point.” The Santalo example was later extended by Hadwiger and Debrunner [1964] to show that even if the convex sets are disjoint the conclusion still may not hold.

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:

THEOREM [Hadwiger 1957]. If B₁,…, B_n is a family of disjoint compact convex sets in the plane with the property that for any 1 ≤ I ≤ j ≤ k ≤ n there is a line meeting each of B_i,B_j,B_k in that order, then there is a line meeting all the sets B_i.