4 convex sets in a plane have a point in common

Question

Let $X_1,X_2,X_3,X_4$ be four sets in the plane such that any three of them have a point in common. Do all four of them have to have a point in common? What if sets are convex?

Attempt: I think all of them should not have a point in common, but I dont how them being convex can influence the answer. Thanks for help.

This works for convex sets: http://en.wikipedia.org/wiki/Helly's_theorem — Beni Bogosel, Nov 18 '13 at 15:29

score 2 · Accepted Answer · answered Nov 18 '13 at 15:29

2

It is easy to come up with a counter-example if the sets are not-convex.

If the sets are convex, read the technique of Helly's Theorem.

answered Nov 18 '13 at 15:29

Calvin Lin

68,864

score 1 · Answer 2 · answered Nov 18 '13 at 15:35

1

As a counter example think of a square $A_1A_2A_3A_4$ and pick $X_i=A_{i-1}A_i \cup A_iA_{i+1}$ (notation modulo $4$).

answered Nov 18 '13 at 15:35

Beni Bogosel

23,381

4 convex sets in a plane have a point in common

2 Answers2