What is a wasserstein barycenter?

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I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. What is it? How is it defined? I can only find very deep measure-theoretic explanations, which are both a bit out of my league as a statistician, who never enjoyed analysis very much, but they simply also lack an intuition behind.