Anti-Vietoris--Rips metric thickenings and Borsuk graphs

For X a metric space and r≥ 0, the anti-Vietoris-Rips metric thickening AVRᵐ(X;r) is the space of all finitely supported probability measures on X whose support has spread at least r, equipped with an optimal transport topology. We study the anti-Vietoris-Rips metric thickenings of spheres. We have a homeomorphism AVRᵐ(Sⁿ;r) Sⁿ for r > π, a homotopy equivalence AVRᵐ(Sⁿ;r) RPⁿ for 2π/3 < r ≤ π, and contractibility AVRᵐ(Sⁿ;r) * for $r=0$. For an n-dimensional compact Riemannian manifold M, we show that the covering dimension of AVRᵐ(M;r) is at most $(n+1)p-1$, where p is the packing number of M at scale r. Hence the k-dimensional {C}ech cohomology of AVRᵐ(M;r) vanishes in all dimensions k≥ (n+1)p. We prove more about the topology of AVRᵐ(Sⁿ;2π/3), which has vanishing cohomology in dimensions $2n+2$ and higher. We explore connections to chromatic numbers of Borsuk graphs, and in particular we prove that for $k>n$, no graph homomorphism Bor(Sᵏ;r) → Bor(Sⁿ;α) exists when α > 2π/3.