Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ 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 $\bigcap_{i=1}^N X_i \neq \varnothing$.

Let $f: \mathbb{R}^d \times \mathbb{R}^n \rightarrow \mathbb{R}$ be such that for all $x \in \mathbb{R}^d$, $y \mapsto f(x,y)$ is convex, and for all $y \in \mathbb{R}^n$, $x \mapsto f(x,y)$ is convex. Suppose $n \leq d$.

Define the sets $$ X_i := \{ x \in \mathbb{R}^d \mid f( x, y_i ) \leq 0 \}, \ \forall i = 1, 2, ..., N,$$ where $y_1, ..., y_N \in \mathbb{R}^n$ are given vectors.

I am wondering if Helly's Theorem holds true with Helly's number $h$ depending on $n$, not on $d$.

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    $\begingroup$ Crossposted on Mathoverflow by "Frank" at mathoverflow.net/questions/133184/… $\endgroup$
    – Boris Bukh
    Jun 9, 2013 at 12:40
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    $\begingroup$ It appears that you have crossposted this question simultaneously. While we don't mind a question being reposted, our site policy prohibits simultaneous crossposting as it duplicates effort and fractures discussion. Crossposting is only permitted after sufficient time has passed and you have not obtained your desired answer elsewhere. When crossposting please summarize the relevant discussions from other sites in your question and link to the copies in both directions. $\endgroup$
    – Kaveh
    Jun 9, 2013 at 21:44
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    $\begingroup$ the MO question has a good answer, I think we should close this one $\endgroup$ Jun 10, 2013 at 0:01
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    $\begingroup$ See math.stackexchange.com/questions/414971/… $\endgroup$ Jun 10, 2013 at 9:05