Helly's theorem and shifts of sets. I

Abstract

The motivation for the considered geometric problems is the study of conditions under which an exponential system is incomplete in spaces of the functions holomorphic in a compact set $C $ and continuous on this compact set. The exponents of this exponential system are zeroes for a sum (finite or infinite) of families of entire functions of exponential type. As $C$ is a convex compact set, this problem happens to be closely connected to Helly's theorem on the intersection of convex sets in the following treatment. Let $C$ and $S $ be two sets in a finite-dimensional Euclidean space being respectively intersections and unions of some subsets. We give criteria for some parallel translation (shift) of set $C$ to cover (respectively, to contain or to intersect) set $S$. These and similar criteria are formulated in terms of geometric, algebraic, and set-theoretic differences of subsets generating $C $ and $S$.