MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 4 added 63 characters in body

This answer provides a nice background to the question.

I can say that a theorem similar to the one in the OP is certainly true. The following was conjectured by Grunbaum and Motzkin in "On components in some families of sets" and later proved by Amenta in "Helly-type theorems and Generalized Linear Programming".

Theorem Let $\mathcal C$ be a family of sets in $\mathbb R^d$ such that the intersection of any non-empty finite subfamily of $\mathcal C$ is the disjoint union of at most $k$ closed convex sets. Then the intersection of all sets in $\mathcal C$ is non-empty if and only if the intersection of any $k(d+1)$ elements of $\mathcal C$ is non-empty.

In particular this implies that the conjecture in the OP is true when $m\geq 2d+3$. I believe there are counter examples for $m\le k(d+1)$, showing it is best possible.
show/hide this revision's text 3 deleted 55 characters in body

In response to the counter-examples given by domotorp,

I can say that a theorem similar to the one in the OP is certainly true. The following was conjectured by Grunbaum and Motzkin in "On components in some families of sets" and later proved by Amenta in "Helly-type theorems and Generalized Linear Programming".

Theorem Let $\mathcal C$ be a family of sets in $\mathbb R^d$ such that the intersection of any non-empty finite subfamily of $\mathcal C$ is the disjoint union of at most $k$ closed convex sets. Then the intersection of all sets in $\mathcal C$ is non-empty if and only if the intersection of any $k(d+1)$ elements of $\mathcal C$ is non-empty.

In particular this implies that the conjecture in the OP is true when $m\geq 2d+3$. I believe there are counter examples for $m\le k(d+1)$, showing it is best possible.
show/hide this revision's text 2 added 172 characters in body

In response to the counter-examples given by domotorp, I can say that a theorem similar to the one in the OP is certainly true. The following was conjectured by Grunbaum and Motzkin in "On components in some families of sets" and later proved by Amenta in "Helly-type theorems and Generalized Linear Programming".

Theorem Let $\mathcal C$ be a family of sets in $\mathbb R^d$ such that the intersection of any non-empty finite subfamily of $\mathcal C$ is the disjoint union of at most $k$ closed convex sets. Then the intersection of all sets in $\mathcal C$ is non-empty if and only if the intersection of any $k(d+1)$ elements of $\mathcal C$ is non-empty.

In particular this implies that the conjecture in the OP is true when $m\geq 2d+3$. I believe there are counter examples for $m\le k(d+1)$, showing it is best possible.
show/hide this revision's text 1