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Math. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We introduce a weak notion of barycenter of a probability measure $\\mu$ on a metric measure space $(X, d, {\\bf m})$, with the metric $d$ and reference measure ${\\bf m}$. Under the assumption that all optimal transport plans transporting ${\\bf m}$ to any probability measure $\\nu$ on $X$ are induced by mappings, we prove that our barycenter $B(\\mu)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $\\mu$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P_2(X)$, pushing a given measure $\\mu$ forward to a measure on $P_2(X)$. We then regularize the solution to the latter problem by the squared Wasserstein distance to the reference measure ${\\bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $\\mu$. 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