Let A ⊂ R2 be a fixed convex set and let X1, . . . , Xn ⊂ R2 be any convex sets such that every three of them intersect a translation of A. Then there exists a translation of A that intersects all sets Xi.
I understood the first two part of the question but I am not able to derive the last part or rather how to prove it. Can someone please help me on how to solve this?
Denote by cm(W) the centre of mass of the set W. For every i, define Yi in such a way that Xi ∩ A' is not equal to φ if and only if cm(A')∈ Yi, for every translation A' of A. Now using Helly’s Theorem we have a point z ∈ Y1 ∩ ... ∩ Yn, we can find a translation A" such that cm(A") = z. By definition of the sets Yi, the set A" intersects all sets Xi