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Borsuk’s conjecture, Ryshkov obstruction, interpolation, Chebyshev approximation, transversal Tverberg’s theorem, and problems. (English. Russian original) Zbl 1069.57500

Proc. Steklov Inst. Math. 239, 55-73 (2002); translation from Tr. Mat. Inst. Im. V. A. Steklova 239, 63-82 (2002).
This is a survey concerning embeddings of polyhedra into Euclidean spaces. 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.
For the entire collection see [Zbl 1059.52002].

MSC:

57Q35 Embeddings and immersions in PL-topology
54C25 Embedding
52A35 Helly-type theorems and geometric transversal theory
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
65D15 Algorithms for approximation of functions
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
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