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In this note, we show that the squared Wasserstein distance can be expressed as the average over the sphere of one dimensional Wasserstein distances. We name this new expression for the Wasserstein Distance Radiant Formula. Using this formula, we are able to highlight new connections between the Wasserstein distances and the Sliced Wasserstein distance, an alternative Wasserstein-like distance that is cheaper to compute. [ABSTRACT FROM AUTHOR]

This paper is motivated by the classical theorem due to Hardy and Littlewood which concerns analytic mappings on the unit disk and relates the growth of the derivative with the Hölder continuity. We obtain a version of this result in a very general setting—for regularly oscillating mappings on a metric space equipped with a weight, which is a continuous and positive function, with values in another metric space. As a consequence, we derive the Hardy and Littlewood theorem for analytic mappings on the unit ball of a normed space. [ABSTRACT FROM AUTHOR]

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs. [ABSTRACT FROM AUTHOR]