The Katchalski–Lewis transversal problem for regular polygons

Abstract

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family $${\mathcal{F}}$$ has property $${T-m}$$ , if there exists a subfamily $${\mathcal{G} \subset \mathcal{F}}$$ with $${|\mathcal{F} - \mathcal{G}| \le m}$$ admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if $${\mathcal{F}}$$ is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which $${\mathcal{F}}$$ has property $${T-m}$$ . In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon $${(n \ge 5)}$$ . We find out that, for $${5 \le n \le 34}$$ , the family has property $${T - 3}$$ ; for $${n \ge 35}$$ , the family has property $${T - 2}$$ .