Intersections of staircase convex sets in ${\mathbb {R}}^3$ and ${\mathbb {R}}^d$

Abstract

We establish the following Helly-type theorems: Let ${\mathcal {K}}$ be a family of sets in ${\mathbb {R}}^3$, and let j be a fixed integer, $0 \le j \le 3$. For every countable subfamily of ${\mathcal {K}}$, assume that the corresponding intersection is consistent relative to staircase paths, is staircase convex, and contains a convex set of dimension at least j. Then $\cap \{K \, : \, K \text { in } {\mathcal {K}} \}$ has these properties as well. For d a fixed integer, $d \ge 1$, define a function f on $\{0,1\}$ by $f(0)=d+1$ , $f(1)=2d$. Let ${\mathcal {K}}$ be a finite family of orthogonal polytopes in ${\mathbb {R}}^d$. For j fixed in $\{ 0,1\}$, assume that every f(j) members of ${\mathcal {K}}$ intersect in a set that is staircase convex and contains a convex subset of dimension at least j. Then $\cap \{K : K \text { in } {\mathcal {K}} \}$ has these properties, too. The number f(j) is best possible for $j=0$ and for $j=1$. There is no analogous Helly number for $2 \le j \le d$.