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 Ufimsk. Mat. Zh., 2014, Volume 6, Issue 3, Pages 98–111 (Mi ufa255)

Helly's theorem and shifts of sets. I

B. N. Khabibullin

Bashkir State University, Z. Validi str., 32, 450074, Ufa, Russia

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 $S$.

Keywords: Helly's theorem, incompleteness of exponential systems, convexity, shift, geometric, algebraic, and set-theoretic differences.

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English version:
Ufa Mathematical Journal, 2014, 6:3, 95–107 (PDF, 432 kB); https://doi.org/10.13108/2014-6-3-95

UDC: 514.17+517.547.2
MSC: 52A35, 52A20

Citation: B. N. Khabibullin, “Helly's theorem and shifts of sets. I”, Ufimsk. Mat. Zh., 6:3 (2014), 98–111; Ufa Math. J., 6:3 (2014), 95–107

Citation in format AMSBIB
\Bibitem{Kha14} \by B.~N.~Khabibullin \paper Helly's theorem and shifts of sets.~I \jour Ufimsk. Mat. Zh. \yr 2014 \vol 6 \issue 3 \pages 98--111 \mathnet{http://mi.mathnet.ru/ufa255} \elib{http://elibrary.ru/item.asp?id=22370786} \transl \jour Ufa Math. J. \yr 2014 \vol 6 \issue 3 \pages 95--107 \crossref{https://doi.org/10.13108/2014-6-3-95} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84928181776} 

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This publication is cited in the following articles:
1. B. N. Khabibullin, “Helly's Theorem and shifts of sets. II. Support function, exponential systems, entire functions”, Ufa Math. J., 6:4 (2014), 122–134
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