I was recently wondering whether there is a weighted version of Helly's theorem.
Helly's theorem tells us that for $n\geq d+1$, a collection of convex subsets of $\mathbb{R}^d$, $\{ A_k \}_{k=1}^n$, satisfying that any $d+1$ sub-collection intersects non-trivially, satisfies that $\cap_{k=1}^n A_k\neq \emptyset$.
I was wondering whether there is a known similar result concerning the measure of the intersection. Namely, are there $N(d)\in \mathbb{N}$ and $c(d)> 0$, such that any collection of convex subsets ,$\{ A_k \}_{k=1}^n$, satisfying
- $n\geq N(d)$
- For every $1\leq k_1<k_2<...<k_{d+1}\leq n$, the Lebesgue measure of $\cap_{\ell=1}^{d+1} A_{k_\ell}$ is at least $\epsilon>0$.
then $Leb(\cap_{k=1}^n A_k)\geq c(d)\cdot \epsilon$? Or perhaps some result in this direction?
I'm asking following this thread, since I have a feeling that some such result should hold. I assume that if this is not true, then probably there are rather easy counterexamples.
Does anyone something about this, or can help resolve the issue?