The dimension of the kernel in an intersection of starshaped sets

We establish the following Helly-type theorem: Let K be a family of compact sets in R-d. If every d + 1 (not necessarily distinct) members of K intersect in a starshaped set whose kernel contains a translate of set A, then boolean AND{K : K in K} also is a starshaped set whose kernel contains a translate of A. An analogous result holds when K is a finite family of closed sets in Rd. Moreover, we have the following planar result: Define function f on {0, 1, 2} by f(0) = f(2) = 3, f(1) = 4. Let K be a finite family of closed sets in the plane. For k = 0, 1, 2, if every f (k) (not necessarily distinct) members of K intersect in a starshaped set whose kernel has dimension at least k, then boolean AND{K : K in K} also is a starshaped set whose kernel has dimension at least k. The number f(k) is best in each case.