2017/03/28 by Kim, Young-Heon, Pass, Brendan
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1703.09754
We introduce a weak notion of barycenter of a probability measure μ on a metric measure space (X, d, \bf m), with the metric d and reference measure \bf m. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter B(μ) is well defined; it is a probability measure on X supported on the set of the usual metric barycenter points of the given measure μ. The definition uses the canonical embedding of the metric space X into its Wasserstein space P(X), pushing a given measure μ forward to a measure on P(X). We then regularize the measure by the Wasserstein distance to the reference measure \bf m, and obtain a uniquely defined measure on X supported on the barycentric points of μ. We investigate various properties of B(μ)