Abstract
The motivation for the considered geometric problems is the study of con- ditions under which an exponential system is incomplete in spaces of the functions holo- morphic in a compact set �� 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 �� 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 �� and �� be two sets in a finite-dimensional Euclidean space being respectively inter- sections and unions of some subsets. We give criteria for some parallel translation (shift) of set �� to cover (respectively, to contain or to intersect) set �� . These and similar criteria are formulated in terms of geometric, algebraic, and set-theoretic differences of subsets generating �� and �� .
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