Geodesic
Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 63-82

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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. 
@article{TM_2002_239_a3,
     author = {S. A. Bogatyi},
     title = {Borsuk's {Conjecture,} {Ryshkov} {Obstruction,} {Interpolation,} {Chebyshev} {Approximation,} {Transversal} {Tverberg's} {Theorem,} and {Problems}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {63--82},
     publisher = {mathdoc},
     volume = {239},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_239_a3/}
}
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S. A. Bogatyi. Borsuk's Conjecture, Ryshkov Obstruction, Interpolation, Chebyshev Approximation, Transversal Tverberg's Theorem, and Problems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 63-82. http://geodesic.mathdoc.fr/item/TM_2002_239_a3/