Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps @math on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps @math . We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.
@cite_36: L eonid Vital’evich Kantorovich (1912–1986) was one of the great mathematicians and economists of the twentieth century. In 2012, the centenary of his birth was marked in St. Petersburg. Short histories were presented describing some of the main parts of his legacy, which continue in importance today: duality in linear programming, the so-called ‘‘MongeKantorovich transportation problem,’’ and the ‘‘Kantorovich metric.’’ Note that 2012 was also the 70th anniversary of the publication of his historic paper on the transport metric. The present article offers a somewhat expanded version ofmy talk on that occasion.
@cite_3: 
@cite_31: Given topological spaces X1, ..., Xn with product space X, probability measures μi on Xi together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces Xi and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems.
@cite_16: Abstract The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W 1 on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved.
@cite_10: Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric and Gaussian inequalities The metric side of optimal transportation A differential point of view on optimal transportation Entropy production and transportation inequalities Problems Bibliography Table of short statements Index.
@cite_12: 
This metric is also called the Kantorovich, optimal transport, or earth mover's metric @cite_36 @cite_10 @cite_12 ; it provides a notion of distance between probability measures defined on a metric space. Although it exists much more generally @cite_16 @cite_31 @cite_3 , the @math -Wasserstein metric on @math can be defined as follows. Given @math with @math and @math , define a @math between @math and @math to be any collection of non-negative real numbers @math such that @math and @math . Define the of the matching @math to be @math . The @math -Wasserstein distance between @math is then the infimum, varying over all matchings @math between @math and @math , of the cost of @math .