In this talk, I will discuss the concept of a barycenter on a general metric space. This notion provides a definition of the “mean” in spaces where the absence of a linear structure makes this concept ambiguous. This notion is particularly useful in Statistics, (metric) geometry, and data science. After providing a general overview, I will focus on the properties of barycenters on what is known as Wasserstein space. The (quadratic) Wasserstein space is the set of probability measures on a given metric space whose second-order moment is finite equipped with a distance coming from optimal mass transport.

Among the questions that naturally arise are the existence, uniqueness, and features of the Wasserstein barycenter as a probability measure. In the final part, I will present recent results obtained in collaboration with J. Ma on barycenters in Wasserstein space over a metric graph.