Ufa Mathematical Journal
    Volume 6, Number 4, pp. 122-134

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

    Khabibullin B.N.


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    Let $\mathcal S$ be a family of sets in $ \mathds{R}^n $, $S$ be the union of all these sets and $C$ be a convex set in $ \mathds{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 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.