Affinely embeddable convex sets (Q579625)

The authors prove the following Helly-type theorem: Let \({\mathcal A}_ r=\{A_ 1\),..., \(A_ r\}\) be a collection of \(r\geq 4\) mutually disjoint closed convex sets in the real projective plane \(P^ 2\) with the property that no element of \({\mathcal A}_ r\) is contained in a convex hull of any other two elements of \({\mathcal A}_ r\). Then \({\mathcal A}_ r\) is affinely embeddable, i.e. there is a line in \(P^ 2\) which does not meet any of the \(A_ i\).