Helly's Theorem and shifts of sets. II. Support function, exponential systems, entire functions

Let $\mathcal S$ be a family of sets in $\mathbb R^n$, $S$ be the union of all these sets and $C$ be a convex set in $\mathbb R^n$. In terms of support functions of sets in $\mathcal S$ and set $C$ we establish necessary and sufficient conditions under which a parallel shift of the set $C$ covers set $S$. We study independently the two-dimensional case, when sets are unbounded, by employing additional characteristics of sets. We give applications of these results to the problems of incompleteness of exponential systems in function spaces.