On a conjecture of Danzer and Grünbaum

Abstract

The main result of the paper is that if A A is a family of homothetic triangles in the plane such that any 9 of them can be pierced by two points, then all members of A A can be pierced by two points. This is best possible in more than one sense: (1) the number 9 cannot be replaced by 8; (2) no similar statement is true for homothetic copies (or even translates) of a symmetric convex hexagon.