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 $\bigcap_{i=1}^N X_i \neq \varnothing$.

If instead $X_1,X_2,...,X_N$ are biconvex sets in $\mathbb{R}^d$, what is "*Helly's number*" $h(d)$ (or an upper bound of it)?

Comment: A set $S \subseteq \mathbb{R}^{n} \times \mathbb{R}^m = \mathbb{R}^d$ is *biconvex* if for all $y \in \mathbb{R}^m$ the set $S_y := \{ x \in \mathbb{R}^n \mid (x,y) \in S \}$ is convex, and for all $x \in \mathbb{R}^n$ the set $S_x := \{ y \in \mathbb{R}^m \mid (x,y) \in S \}$ is convex as well.