# Helly's Theorem for Rectangles

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \varnothing$. Then $\bigcap_{i=1}^N X_i \neq \varnothing$.

In particular, if $X_1,X_2,...,X_N$ are rectangle in $\mathbb{R}^d$, what is "Helly's number" $h(d)$?

Comment: A "rectangle" is a (convex) set of the kind $[ a_1, b_1 ] \ \times\ ... \ \times \ [a_d, b_d]$ .

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The question is unclear in that you define $h(d)$ to be $d+1$ in the question. Did you mean to ask whether and how the bound $h(d)$ can be chosen lower than $d+1$ if the $X_i$ are known to be rectangles and not just convex sets? –  joriki May 30 '13 at 13:06
Let me clarify that $h(d) = d+1$ for general convex sets, see [en.wikipedia.org/wiki/Helly's_theorem]. Yes, I am asking whether $h(d)$ can be chosen smaller than $d+1$ if the $X_i$'s are known to be rectangles. –  Adam May 30 '13 at 13:23