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The Knaster problem and almost spherical sections. (Russian) Zbl 0684.52002

Knaster’s problem: Find point configurations \(A_ 1,...,A_ k\) on the sphere \(S^{n-1}\subset {\mathbb{R}}^ n\), such that for each continuous map f: \(S^{n-1}\to {\mathbb{R}}^ m\), there is a rotation a for which \(f(a(A_ 1))=...=f(a(A_ k))\). Call such a configuration a solution. The author describes various solutions as consequences of the following results.
Theorem 1. If \(A_ 1,...,A_ p\) are vertices of a regular p-gon inscribed in a great circle of \(S^{n-1}\), \(n\geq 3\), and if p is a prime number \(>2n-2,\) then for each continuous function f, \((m=1)\), there are indices \(1\leq i_ 1<...<i_{2n-2}\leq p\) and a rotation a for which the values \(f(a(A_ j))\), \(j=i_ 1,...,i_{2n-2}\), are all equal. If \(p<2n-2,\) the same is true with \(j=1,...,p.\)
Theorem 2. If points \(A_ 1,A_ 2,A_ 3\) of \(S^ 2\) are vertices of an equilateral triangle and f, g are continuous maps of \(S^ 2\) into \({\mathbb{R}}^ 2\), then there is a rotation a such that the points \(f(a(A_ j))\), \(j=1,2,3\) are equal and \(g(a(A_ 1))=g(a(A_ 2)).\)
As further applications, he studies an infinitesimal version of Knaster’s problem as well as proving some asphericity results.
Reviewer: W.J.Firey

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A55 Spherical and hyperbolic convexity
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