Gaussian smoothed sliced Wasserstein distance has been recently introduced\nfor comparing probability distributions, while preserving privacy on the data.\nIt has been shown, in applications such as domain adaptation, to provide\nperformances similar to its non-private (non-smoothed) counterpart. However,\nthe computational and statistical properties of such a metric is not yet been\nwell-established. In this paper, we analyze the theoretical properties of this\ndistance as well as those of generalized versions denoted as Gaussian smoothed\nsliced divergences. We show that smoothing and slicing preserve the metric\nproperty and the weak topology. We also provide results on the sample\ncomplexity of such divergences. Since, the privacy level depends on the amount\nof Gaussian smoothing, we analyze the impact of this parameter on the\ndivergence. We support our theoretical findings with empirical studies of\nGaussian smoothed and sliced version of Wassertein distance, Sinkhorn\ndivergence and maximum mean discrepancy (MMD). In the context of\nprivacy-preserving domain adaptation, we confirm that those Gaussian smoothed\nsliced Wasserstein and MMD divergences perform very well while ensuring data\nprivacy.\n