Heterochromatic Higher Order Transversals for Convex Sets

Let $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be an infinite collection of families of compact convex sets in $\mathbb{R}^{d}$. An infinite sequence of compact convex sets $\{B_n\}_{n\in\mathbb{N}}$ is said to be a heterochromatic sequence with respect to $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ if each $B_n$ comes from a different family $\mathcal{F}_{i_n}$, and $\{B_n\}_{n\in\mathbb{N}}$ is said to be a strongly heterochromatic sequence of $\left\{ \mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ if $\forall n\in\mathbb{N}, \; B_n \in \mathcal{F}_n$. We show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families compact $(r, R)$-fat convex sets in $\mathbb{R}^{d}$ and if every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $k+2$ convex sets that can be pierced by a $k$-flat then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by finitely many $k$-flats. Additionally, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families of compact convex sets in $\mathbb{R}^{d}$ where each $\mathcal{F}_{n}$ is a family of closed balls (axis parallel boxes) in $\mathbb{R}^{d}$ and every (strongly) heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $2$ intersecting closed balls (boxes) then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by a finite number of points from $\mathbb{R}^{d}$. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets.