Helly-type problems for convex quadrilaterals in the plane

A family of sets in the plane is said to have a 3-transversal if there exists a set of 3 points such that each member of the family contains at least one of them. A conjecture of Grünbaum’s says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [1]). We provide a straightforward proof for the conjecture for the family of translates of a closed convex quadrilateral without parallel sides in the plane. Moreover, in our proof we obtain exactly the 3-transversals, i.e. the concrete 3-point sets the conjecture claims. The proof is valid for some other convex polygons and it is likely that we can prove the conjecture in the same straightforward way. For brevity’s sake, a family of sets is said to be Π, or to have a 3-transversal if there exists a set of 3 points such that each member of the family contains at least one of them. The family is said to be Π2 if every two sets of the family have a nonempty intersection. Grünbaum’s conjecture says Conjecture For a family of translates of a compact convex set in the plane, Π2 implies Π. In a recent paper by M. Katchalski and D. Nastir (see [2]) the above conjecture of Grünbaum was mentioned again. Karasev [1] gives an affirmative answer to the conjecture. We provide a straightforward proof for the conjecture for the case of a quadrilateral instead of a general compact convex set. In the same way we can prove the conjecture for triangles, parallelograms and trapezoids, etc. Accordingly, it is ∗ This research was supported by NSFH and SFHEM 210 LIPING YUAN AND REN DING