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Metric thickenings, Borsuk-Ulam theorems, and orbitopes. (English) Zbl 1443.05188

Let \(M\) be a metric space, \(r\geq 0\) and \(X \subset M\). Very recently, in [M. Adamaszek et al., SIAM J. Appl. Algebra Geom. 2, No. 4, 597–619 (2018; Zbl 1406.53045)] there was introduced the notion of Vietoris-Rips thickening \(VR^m(X;r)\) which captures local geometric properties of the space better than the standard construction of the Vietoris-Rips simplicial complex \(VR(X;r)\) in the case when \(X\) is not locally finite and so the Vietoris-Rips simplicial complex is not metrizable. That gives the possibility to describe the homotopy type of the Vietoris-Rips thickening of the \(n\)-sphere. The main goal of this paper is to relate this new construction of the metric thickenings of the circle (and more generally then \(n\)-sphere) to convexity properties of orbits of circle actions on Euclidean space, to Borsuk-Ulam type theorems, and to the structure of zeros of trigonometric polynomials with a prescribed spectrum.

MSC:

05E45 Combinatorial aspects of simplicial complexes
52B15 Symmetry properties of polytopes
54E35 Metric spaces, metrizability
55P10 Homotopy equivalences in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology

Citations:

Zbl 1406.53045
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References:

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