<p>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 $\bigcap_{i=1}^N X_i \neq \varnothing$.</p>

<p>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 $n \leq d$.</p>

<p>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.</p>

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

CC BY-SA 3.0
2013-06-09T12:40:24.957Z
Crossposted on Mathoverflow by "Frank" at http://mathoverflow.net/questions/133184/hellys-number-from-biconvex-functions
Boris Bukh
CC BY-SA 3.0
2013-06-09T21:44:58.177Z
It appears that you have crossposted this question *simultaneously*. While we don't mind a question being reposted, our [site policy](http://meta.cstheory.stackexchange.com/a/231/) 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.
Kaveh
CC BY-SA 3.0
2013-06-10T00:01:26.290Z
the MO question has a good answer, I think we should close this one
Sasho Nikolov
CC BY-SA 3.0
2013-06-10T09:05:06.780Z
See http://math.stackexchange.com/questions/414971/hellys-number-from-biconvex-functions
András Salamon
<co.combinatorics><cg.comp-geom><convex-geometry>
Helly's number from biconvex functions
user693
# Helly's number from biconvex functions

(Asked by: username_0 on 2013-06-08)

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 $d$.

**Tags:** co.combinatorics, cg.comp-geom, convex-geometry
> **Comments:**
> **username_1**: Crossposted on Mathoverflow by "Frank" at http://mathoverflow.net/questions/133184/hellys-number-from-biconvex-functions  
> **username_2**: It appears that you have crossposted this question *simultaneously*. While we don't mind a question being reposted, our [site policy](http://meta.cstheory.stackexchange.com/a/231/) 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.  
> **username_3**: the MO question has a good answer, I think we should close this one  
> **username_4**: See http://math.stackexchange.com/questions/414971/hellys-number-from-biconvex-functions  

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