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 Tr. Mat. Inst. Steklova, 2002, Volume 239, Pages 63–82 (Mi tm359)

Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems

S. A. Bogatyi

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: S. S. Ryshkov's solution of K. Borsuk's problem about $k$-regular embeddings is discussed. The results of Haar, Kolmogorov, and Rubinshtein are presented concerning the relation between $k$-regular mappings and interpolation, the number of zeros, and the low-dimensionality of the polyhedron of best Chebyshev approximations. The Tverberg transversal theorem is proved, and the place of the colored Tverberg theorem in the class of the problems discussed is highlighted. Many unsolved problems are formulated.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 239, 55–73

Bibliographic databases:
UDC: 515.127.15

Citation: S. A. Bogatyi, “Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems”, Discrete geometry and geometry of numbers, Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov, Tr. Mat. Inst. Steklova, 239, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 63–82; Proc. Steklov Inst. Math., 239 (2002), 55–73

Citation in format AMSBIB
\Bibitem{Bog02}
\by S.~A.~Bogatyi
\paper Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems
\inbook Discrete geometry and geometry of numbers
\bookinfo Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 239
\pages 63--82
\publ Nauka, MAIK «Nauka/Inteperiodika»
\mathnet{http://mi.mathnet.ru/tm359}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1975135}
\zmath{https://zbmath.org/?q=an:1069.57500}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 239
\pages 55--73

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This publication is cited in the following articles:
1. A. Yu. Volovikov, “The genus of $G$-spaces and topological lower bounds for chromatic numbers of hypergraphs”, J. Math. Sci., 144:5 (2007), 4387–4397
2. A. Yu. Volovikov, “Coincidence points of maps of $\mathbb Z_p^n$-spaces”, Izv. Math., 69:5 (2005), 913–962
3. S. A. Bogatyi, V. M. Valov, “Roberts-type embeddings and conversion of transversal Tverberg's theorem”, Sb. Math., 196:11 (2005), 1585–1603
4. Bogatyi S.A., “Finite–to–one maps”, Topology and Its Applications, 155:17–18 (2008), 1876–1887
5. Arocha J.L., Bracho J., Montejano L., Alfonsin J.L.R., “Transversals to the convex hulls of all k-sets of discrete subsets of R-n”, J Combin Theory Ser A, 118:1 (2011), 197–207
6. S. I. Bogataya, S. A. Bogatyi, E. A. Kudryavtseva, “An inverse theorem on ‘economic’ maps”, Sb. Math., 203:4 (2012), 554–568
7. K. I. Oblakov, T. A. Oblakova, “Embeddings of graphs into Euclidean space under which the number of points that belong to a hyperplane is minimal”, Sb. Math., 203:10 (2012), 1518–1533
8. S. A. Bogatyi, “Generic planes conjecture”, Moscow University Mathematics Bulletin, 67:5-6 (2012), 200–205
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