Type: Article

A question from a famous paper of Erdos

Barany, Imre; Roldán-Pensado, Edgardo;

Abstract:

Let K be a planar convex body and let N(K)?N?? denote the smallest value for which there exists a point p??K such that every circle with center p intersects ?K in at most N(K) points. Paul Erd?s conjectured that N(K)?2 for every planar convex body. This conjecture is false since N(T)=4 for an acute triangle T. Furthermore, the authors show that there exists a planar convex body C with N(C)=6. They conjecture that N(K) is bounded by a finite constant independent of K and remark that this constant is probably 6. They also prove that N(K)<? for every planar convex body K.