$k$-partitions and a characterization for compact unions of $k$ starshaped sets

Abstract:

For natural numbers $k$ and $j$, define $\sigma \left ( {1,j} \right ) = j + 2,\;\sigma \left ( {2,1} \right ) = 6$, and $\sigma \left ( {k,j} \right ) = {k^2}j + 1$ otherwise. The set $S$ in ${R^d}$ has property $A\left ( k \right )$ if and only if $S$ is a finite union of one-dimensional convex sets and for every $\sigma \left ( {k,1} \right )$-member subset $F$ of $S$ there correspond points ${c_1}, \ldots ,{c_k}$ (depending on $F$) such that each point of $F$ sees via $S$ some ${c_i},1 \leqslant i \leqslant k$. The following results are established. (1) Let $k$ and $j$ be fixed natural numbers, and let $\mathcal {L}$ be a collection of sets such that every $j + 1$ members meet in at most one point. Then $\mathcal {L}$ has a $k$-partition ${\mathcal {L}_1}, \ldots ,{\mathcal {L}_k}$ with $\cap \left \{ {L:L\;{\text {in}}\;{\mathcal {L}_i}} \right \} \ne \emptyset$ if and only if every $\sigma \left ( {k,j} \right )$ or fewer members of $\mathcal {L}$ have such a $k$-partition. (2) Let $S$ be compact in ${R^d}$. The set $S$ is a union of $k$ starshaped sets if and only if there is a sequence of compact sets $\left \{ {{S_i}} \right \}$ converging to $S$ (relative to the Hausdorff metric) such that each set ${S_i}$ satisfies property $A\left ( k \right )$. The first result yields a piercing number for certain families of sets, while the second provides a characterization for compact unions of $k$ starshaped sets in ${R^d}$.