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On F-convexity and related problems. Yuan, Liping


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. In 2007, Heppes 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 talk, 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 $T - 2$. This is a joint work with Qingdan Du and Tudor Zamfirescu.

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