A variant of Helly’s theorem

Abstract

1. Helly's [8] theorem on intersections of convex sets (If every k+1 members of a family of compact, convex subsets of Ek have a nonempty intersection, then the intersection of all the members of the family is not empty) has been generalized in various directions. Helly himself gave (in [9]) a generalization to families of not necessarily convex sets, in which the intersections of any 2, 3, * * *, k members are assumed to satisfy certain conditions (which are automatically fulfilled for families of convex sets). In other papers (e.g., [3; 7; 13 ]) problems related to Helly's theorem were considered under weaker assumptions on the of sets with nonempty intersections, but restricted to families consisting of translates (or of homothetic images) of one convex set. Similar families have been considered also in connection with theorems of Helly's type for common transversals instead of common points (a list of references is given in [4]). On the other hand, it was shown [2] that if families of affine transforms of one set are considered, the convexity of the sets is necessary for the validity of Helly's theorem in its original form. The present paper results from an attempt to find whether there exists some theorem of Helly's type for nonconvex sets, in case only families consisting of affine or other appropriate transforms of one set are considered and, possibly, additional conditions imposed. But, as is easily seen by a slight modification of the example on p. 70 of [5] (or by Example 1 of the present paper), even if only families of translates are considered, and the sets assumed to consist of only two convex components, there exists no critical number corresponding to k+1 in Helly's theorem. Nevertheless, the following individual theorem is valid and, as shown by the examples in ?4, in many respects the best possible.