This paper introduces a notion of barycenter in the Wasserstein space, which generalizes McCann's interpolation to the case of more than two measures. The authors provide existence, uniqueness, characterizations, and regularity results for the barycenter, and relate it to the multimarginal optimal transport problem considered by Gangbo and Święch. They also consider some examples and rigorously solve the Gaussian case. The paper discusses convexity of functionals in the Wasserstein space. The barycenter is defined as the minimizer of an averaged squared Wasserstein distance. The authors derive a dual problem and show that the barycenter problem is equivalent to a multi-marginal optimal transport problem. They also establish an $ L^\infty $ regularity result for the barycenter and discuss convexity properties of functionals in the Wasserstein space. The paper concludes with examples, including the case of one-dimensional space and the case of two measures.This paper introduces a notion of barycenter in the Wasserstein space, which generalizes McCann's interpolation to the case of more than two measures. The authors provide existence, uniqueness, characterizations, and regularity results for the barycenter, and relate it to the multimarginal optimal transport problem considered by Gangbo and Święch. They also consider some examples and rigorously solve the Gaussian case. The paper discusses convexity of functionals in the Wasserstein space. The barycenter is defined as the minimizer of an averaged squared Wasserstein distance. The authors derive a dual problem and show that the barycenter problem is equivalent to a multi-marginal optimal transport problem. They also establish an $ L^\infty $ regularity result for the barycenter and discuss convexity properties of functionals in the Wasserstein space. The paper concludes with examples, including the case of one-dimensional space and the case of two measures.