The Hadwiger transversal theorem for pseudolines SAUGATA BASU School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 USA e-mail: saugata@math.gatech.edu JACOB E. GOODMAN † Department of Mathematics City College, CUNY New York, NY 10031 USA e-mail: jegcc@cunyvm.cuny.edu ANDREAS HOLMSEN ‡ Department of Mathematics University of Bergen Johannes Brunsgt. 12, 5008 Bergen Norway e-mail: andreash@mi.uib.no RICHARD POLLACK § Courant Institute, NYU 251 Mercer St. New York, NY 10012 USA e-mail: pollack@courant.nyu.edu June 11, 2004 Abstract 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. In 1940 Santal´ o showed [12], by an example, that Vincensini’s proof [13] of an extension of Helly’s theorem was incorrect. Vincensini claimed 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 line 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 Santal´ o example was later extended by Hadwiger and Debrunner [9] 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: Supported in part by NSF grant CCR-0049070 and NSF Career Award 0133597. † Supported in part by NSA grant MDA904-03-I-0087 and PSC-CUNY grant 65440-0034. ‡ Supported in part by the Mathematical Sciences Research Institute, Berkeley. § Supported in part by NSF grant CCR-9732101. 1