Helly-type Theorems for Infinite and for Finite Intersections of Sets Starshaped via Staircase Paths

Let d be a fixed integer, 0 ≤ d ≤ 2, and let K be a family of simply connected sets in the plane. For every countable subfamily {Kn : n ≥ 1} of K, assume that ∩{Kn : n ≥ 1} is starshaped via staircase paths and that its staircase kernel contains a convex set of dimension at least d. Then ∩{K : K in K} has these properties as well. For the finite case, define function f on {0, 1} by f(0) = 3, f(1) = 4. Let K = {Ki : 1 ≤ i ≤ n} be a finite family of compact sets in the plane, each having connected complement. For d fixed, d {0, 1}, and for every f(d) members of K, assume that the corresponding intersection is starshaped via staircase paths and that its staircase kernel contains a convex set of dimension at least d. Then ∩{Ki : 1 ≤ i ≤ n} has these properties, also. There is no analogous Helly number for the case in which d = 2. Each of the results above is best possible. MSC 2000: 52.A30, 52.A35