Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that combines a concept of ‘Tverberg unavoidable subcomplexes’ with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space.
Thus, we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljević and Vrećica (‘The colored Tverberg's problem and complexes of injective functions’, J. Combin. Theory, Ser. A 61 (1992) 309–318). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (‘A generalized van Kampen–Flores theorem’, Proc. Amer. Math. Soc. 11 (1991) 559–565) and Volovikov (‘On the van Kampen–Flores theorem’, Math. Notes (5) 59 (1996) 477–481), an affine version of their ‘j‐wise disjoint’ Tverberg theorem, and a topological version of Soberón's result (‘Equal coefficients and tolerance in coloured Tverberg partitions’, Proceeding of the 29th Annual Symposium on Computational Geometry (SoCG) (ACM, Rio de Janeiro, 2013), 91–96) on Tverberg points with equal barycentric coordinates.
Given a sample of points X in a metric space M and a scale r > 0, the Vietoris-Rips simplicial complex VR(X; r) is a standard construction to attempt to recover M from X up to homotopy type. A deficiency of this approach is that VR(X; r) is not metrizable if it is not locally finite, and thus does not recover metric information about M . We attempt to remedy this shortcoming by defining a metric space thickening of X, which we call the Vietoris-Rips thickening VR m (X; r), via the theory of optimal transport. When M is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem (VR m (M ; r) ≃ M for r sufficiently small) with a simpler proof: homotopy equivalence VR m (M ; r) → M is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map M ֒→ VR m (M ; r). Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the n-sphere at the first positive scale parameter r where the homotopy type changes.
We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions.We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. Moreover, our results extend the work of Musin on quantitative Sperner-type results for PL manifolds.
In 1960 Grünbaum asked whether for any finite mass in R d there are d hyperplanes that cut it into 2 d equal parts. This was proved by Hadwiger (1966) for d ≤ 3, but disproved by Avis (1984) for d ≥ 5, while the case d = 4 remained open.
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