This article is cited in 1 scientific paper (total in 1 paper)
Helly's theorem and shifts of sets. I
B. N. Khabibullin
Bashkir State University, Z. Validi str., 32, 450074, Ufa, Russia
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 $S$.
Helly's theorem, incompleteness of exponential systems, convexity, shift, geometric, algebraic, and set-theoretic differences.
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Ufa Mathematical Journal, 2014, 6:3, 95–107 (PDF, 432 kB); https://doi.org/10.13108/2014-6-3-95
MSC: 52A35, 52A20
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
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\paper Helly's theorem and shifts of sets.~I
\jour Ufimsk. Mat. Zh.
\jour Ufa Math. J.
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This publication is cited in the following articles:
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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