Type: Article

Line transversals to translates of a convex body

Jerónimo-Castro, J.; Roldán-Pensado, E.;

Abstract:

Let K be a planar convex body, and F={xi+K:i?I} be a finite family of translates of K. The family F is called a T-family if there exists a line intersecting all members of F. For n?N the family F is said to be a T(n)-family if every subfamily of F with n members is a T-family. Let ?(K,n) be the smallest positive number such that for any T(n)-family F of translates of K the family ?(K,n)F is a T-family. Note that for ?>0 the family ?F is defined as {xi+?K:i?I}. The following facts are known: (i) ?(K,3)?2 if K is an arbitrary convex body, and ?(Q,3)=2 if Q is the unit square [J. Eckhoff, Transversalenprobleme vom Gallai schen Typ, doctoral dissertation, Georg-August-Univ. Göttingen, 1969]; (ii) ?(B,4)=1+5?2 if B is the unit disc [J. Jerónimo Castro, Discrete Comput. Geom. 37 (2007), no. 3, 409–417; MR2301526]. In the paper under review it is proved that (a) ?(K,3)=2 if and only if K is parallelogram, and (b) ?(K,4)?1+5?2 for an arbitrary planar convex body. Note that statement (a) was observed by Eckhoff in his dissertation, but according to the reviewer's knowledge a proof has never been published. Regarding statement (b) the authors notice that equality is possible not only for the unit disc; see (ii). For the proof of the above statements the following interesting lemma is used: If K1 and K2 are centrally symmetric convex bodies, then ?(K1,k)??(K2,k)??(K1,K2), where ?(K1,K2) is the Banach-Mazur distance between K1 and K2.