LAUSR

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 $${\mathcal{F}}$$F has property $${T-m}$$T-m, if there exists a subfamily $${\mathcal{G} \subset \mathcal{F}}$$G⊂F with $${|\mathcal{F} - \mathcal{G}| \le m}$$|F-G|≤m admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if $${\mathcal{F}}$$F is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which $${\mathcal{F}}$$F has property $${T-m}$$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 $${(n \ge 5)}$$(n≥5). We find out that, for $${5 \le n \le 34}$$5≤n≤34, the family has property $${T - 3}$$T-3 ; for $${n \ge 35}$$n≥35, the family has property $${T - 2}$$T-2.