This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Wasserstein space. Replacing the Wasserstein original metric by a sliced approximation over 1D distributions allows us to use a fast stochastic gradient descent algorithm. This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multiscale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures.