Dual theorems on central points and their generalizations
Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1459-1479 Cet article a éte moissonné depuis la source Math-Net.Ru

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Analogues of theorems on a central point, a central transversal and also of Tverberg's theorem are proved in the context when arrangements of hyperplanes or planes of fixed dimension are considered in place of point sets. Bibliography: 20 titles.
Keywords: center point theorem, Tverberg's theorem, Brower's fixed point theorem.
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R. N. Karasev. Dual theorems on central points and their generalizations. Sbornik. Mathematics, Tome 199 (2008) no. 10, pp. 1459-1479. http://geodesic.mathdoc.fr/item/SM_2008_199_10_a2/

[1] B. H. Neumann, “On an invariant of plane regions and mass distributions”, J. London Math. Soc., 20:4 (1945), 226–237 | DOI | MR | Zbl

[2] R. Rado, “A theorem on general measure”, J. London Math. Soc., 21:4 (1946), 291–300 | DOI | MR | Zbl

[3] B. Grünbaum, “Partitions of mass-distributions and of convex bodies by hyperplanes”, Pacific J. Math., 10:4 (1960), 1257–1261 | MR | Zbl

[4] J. Eckhoff, “Helly, Radon, and Carathéodory type theorems”, Handbook of convex geometry, North-Holland, Amsterdam, 1993, 389–448 | MR | Zbl

[5] L. Danzer, B. Grünbaum, V. Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math., 7, Amer. Math. Soc., Providence, RI, 1963 | MR | MR | Zbl | Zbl

[6] H. Tverberg, “A generalization of Radon's theorem”, J. London Math. Soc., 41:1 (1966), 123–128 | DOI | MR | Zbl

[7] R. T. Zivaljević, S. T. Vrećica, “An extension of the ham sandwich theorem”, Bull. London Math. Soc., 22:2 (1990), 183–186 | DOI | MR | Zbl

[8] V. L. Dolnikov, “O razbienii sistemy mer podprostranstvom mnogochlenov”, Konstruirovanie algoritmov i reshenie zadach matematicheskoi fiziki. Doklady 8 Vsesoyuznogo seminara “Teoreticheskie osnovy i konstruirovanie chislennykh algoritmov resheniya zadach matematicheskoi fiziki” (Krasnovidovo, 1990), IPM im. Keldysha, M., 1991, 80–85

[9] V. L. Dol'nikov, “A generalization of the ham sandwich theorem”, Math. Notes, 52:2 (1992), 771–779 | DOI | MR | Zbl | Zbl

[10] S. T. Vrećica, R. T. Živaljević, “New cases of the colored Tverberg theorem”, Jerusalem combinatorics'93 (Jerusalem, Israel, 1993), Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994, 325–334 | MR | Zbl

[11] A. Yu. Volovikov, “On a topological generalization of the Tverberg theorem”, Math. Notes, 59:3 (1996), 324–326 | DOI | MR | Zbl

[12] G. E. Shilov, B. L. Gurevich, Integral, measure and derivative: a unified approach, Prentice-Hall, Englewood Cliffs, NJ, 1966 | MR | MR | Zbl | Zbl

[13] V. L. Dol'nikov, “Generalized transversals of families of sets in $\mathbb R^n$ and connections between the Helly and Borsuk theorems”, Soviet Math. Dokl., 36:3 (1988), 519–522 | MR | Zbl

[14] G. Luke, A. S. Mishchenko, Vector bundles and their applications, Math. Appl., 447, Kluwer Acad. Publ., Dordrecht, 1998 | MR | MR | Zbl | Zbl

[15] J. W. Milnor, J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., 76, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1974 | MR | MR | Zbl | Zbl

[16] M. A. Kervaire, “Relative characteristic classes”, Amer. J. Math., 79:3 (1957), 517–558 | DOI | MR | Zbl

[17] E. S. Polovinkin, Elementy teorii mnogoznachnykh otobrazhenii, Moskovskii fiziko-tekhnicheskii institut, M., 1982

[18] J. Matoušek, Using the Borsuk–Ulam theorem, Lectures on topological methods in combinatorics and geometry, Universitext, Springer-Verlag, Berlin, 2003 | MR | Zbl

[19] W. Yi Hsiang, Cohomology theory of topological transformation groups, Ergeb. Math. Grenzgeb., 85, Springer-Verlag, New-York–Heidelberg, 1975 ; U. I. Syan, Kogomologicheskaya teoriya topologicheskikh grupp preobrazovanii, Mir, M., 1979 | MR | Zbl | MR | Zbl

[20] I. Bárány, D. G. Larman, “A colored version of Tverberg's theorem”, J. London Math. Soc. (2), 45:2 (1992), 314–320 | DOI | MR | Zbl