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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 compact convex 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".

3 added 9 characters in body

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 convex 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".

2 added 2 characters in body

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 ${\cal F}=P_,P_2,\dots,P_m$ 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$is also the disjoint union of two convex sets. Then the intersections of all members of$\cal F$is not empty. Remark: Micha A. Perles showed (in the 70s) that even when$d=2\$ you cannot replace "two" by "48".

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