Here is a curious conjectural extension of Helly's theorem.

It may follow (if true) from a useful theorem of the kind asked in this MO question:

Conjecture:Let ${\cal F}=P_1,P_2,\dots,P_m$ be a family all whose members are disjoint union of two convex sets in $R^d$. Suppose also that(1) $m \ge d+2$

(2) Every intersection of $i$ members of $\cal F$, $i < m$ is also the disjoint union of two NONEMPTY

compactconvex sets.Then the intersections of all members of $\cal F$ is not empty.

Remark: Micha A. Perles showed (in the 70s) that even when $d=2$ you cannot replace "two" by "48".