Geodesic
Helly's theorem and shifts of sets. I
Ufa mathematical journal, Tome 6 (2014) no. 3, pp. 95-107 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, and set-theoretic differences.
Mots-clés : algebraic 
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B. N. Khabibullin. Helly's theorem and shifts of sets. I. Ufa mathematical journal, Tome 6 (2014) no. 3, pp. 95-107. http://geodesic.mathdoc.fr/item/UFA_2014_6_3_a6/

[1] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Fizmatgiz, M., 1956

[2] B. Ya. Levin, Lectures on entire functions, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[3] Khabibullin B. N., Polnota sistem eksponent i mnozhestva edinstvennosti, izdanie chetvertoe, dopolnennoe, RITs BashGU, Ufa, 2012

[4] Khabibullin B. N., “Translyaty vypuklykh mnozhestv”, Sovremennye metody teorii kraevykh zadach, Materialy Voronezhskoi vesennei matematicheskai shkoly “Pontryaginskie chteniya–XXIV”, Izdatelsko-poligraficheskii tsentr VGU, Voronezh, 2013, 207–208

[5] Khabibullin B. N., “Teorema Khelli, translyaty mnozhestv i opornaya funktsiya”, Nelineinye uravneniya i kompleksnyi analiz, Sbornik tezisov, Institut matematiki UNTs RAN, Ufa, 2013, 51–53

[6] Khabibullin B. N., “Teorema Khelli i pokrytie translyatami”, Materialy XI Kazanskaya letnyaya shkola-konferentsiya “Teoriya funktsii, ee prilozheniya i smezhnye voprosy”, Institut matematiki i mekhaniki im. N. I. Lobachevskogo, Kazan, 2013, 196–199

[7] B. N. Khabibullin, “Helly's Theorem and translation of convex sets”, Asymptotic geometric analysis, EIMI, Saint-Petersburg, 2013, 9–10

[8] Dantser L., Gryunbaum B., Kli V., Teorema Khelli, Mir, M., 1968

[9] Rokafellar R. T., Vypuklyi analiz, Mir, M., 1973

[10] Leikhtveis K., Vypuklye mnozhestva, Nauka, M., 1985 | MR

[11] Tikhomirov V. M., “Vypuklyi analiz”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 14, Moskva, 1987, 5–101 | MR | Zbl

[12] Magaril-Ilyaev G. G., Tikhomirov V. M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[13] Polovinkin E. S., Balashov M. V., Elementy vypuklogo i silno vypuklogo analiza, Fizmatlit, M., 2004

[14] R. T. Rockafellar, “Helly's theorem and minima of convex functions”, Duke Math. J., 32 (1965), 381–398 | DOI | MR

[15] L. Sandgren, “On convex cones”, Math. Scand., 2 (1954), 19–28 | MR | Zbl

[16] V. Klee, “Infinite-dimensional intersection theorems”, Convexity, Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, ed. V. Klee, 1963, 349–360 | DOI | MR | Zbl

[17] Pontryagin L. S., “Lineinye differentsialnye igry presledovaniya”, DAN SSSR, 175 (1967), 764–766 | Zbl

[18] Pontryagin L. S., “Lineinye differentsialnye igry presledovaniya”, Matem. sbornik, 112(154):3(7) (1980), 307–330 | MR | Zbl

[19] Petrov N. N., Vvedenie v vypuklyi analiz, Udmurtskii gosudarstvennyi universitet, Izhevsk, 2009

[20] Tolstonogov A. A., Differentsialnye vklyucheniya v banakhovom prostranstve, Nauka, M., 1986 | MR | Zbl

[21] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1995 | MR

[22] Pecherskii S. L., “Znachenie Shepli TP igr, raznosti $c$-yader vypuklykh igr i tochka Shteinera vypuklykh kompaktov”, Matematicheskaya teoriya igr i eë prilozheniya, 4:3 (2012), 58–85

[23] Avvakumov S. N., Kiselev Yu. N., “Opornye funktsii nekotorykh spetsialnykh mnozhestv, konstruktivnye protsedury sglazhivaniya, geometricheskaya raznost”, Problemy dinamicheskogo upravleniya, 1, MAKS Press, M., 2005, 24–110 elektronnaya versiya: http://oc.cs.msu.su/download/76/kiselev05.pdf | MR

[24] Dolnikov V. L., Teoremy tipa Khelli dlya transversalei semeistv mnozhestv i ikh prilozheniya, Dissertatsiya i avtoreferat na soiskanie uchenoi stepeni d.f.-m.n., Yaroslavl, 2000

[25] Bogatyi S. A., “Topologicheskaya teorema Khelli”, Fundament. i prikl. matem., 8:2 (2002), 365–405 | MR | Zbl

[26] Bobylev N. A., “Teorema Khelli dlya zvëzdnykh mnozhestv”, Trudy mezhdunarodnoi konferentsii, posvyaschennoi 90-letiyu so dnya rozhdeniya L. S. Pontryagina (Moskva, 31 avgusta – 6 sentyabrya 1998 g.), v. 7, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obzory, 68, Geometriya i topologiya, VINITI, M., 1999, 16–26 | MR | Zbl

[27] Karasev R. N., “Topologicheskie metody v kombinatornoi geometrii”, UMN, 63:6(384) (2008), 39–90 | DOI | MR | Zbl