We prove the following Helly-type result.
Let $\mathcal{C}_1,\ldots,\mathcal{C}_{3d}$ be finite families of convex bodies
in $\mathbb{R}^d$. Assume that for any colorful choice of $2d$ sets, $C_{i_k}\in
\mathcal{C}_{i_k}$ for each $1\leq k\leq 2d$ with
$1\leq i_1<\ldots<i_{2d}\leq 3d$, the intersection
$\bigcap\limits_{k=1}^{2d} C_{i_k}$ contains an ellipsoid of volume at least 1.
Then there is a color class $1\leq i \leq 3d$ such that $\bigcap\limits_{C\in \C_i} C$
contains an ellipsoid of volume at least $d^{-O(d^2)}$.
Joint work with Gábor Damásdi and Viktória Földvári.