The notion of metric privacy, introduced in 2013, is an elegant notion of privacy for general metric spaces. It generalizes the (1) classical differential privacy, (2) edge privacy for graphs and (3) geo-indistinguishability for location privacy. In this talk, I study the minimax accuracy problem for metric privacy and show that in certain cases, this problem reduces to estimating two geometric quantities, which I call the "entropic scale" and "diametric scale". I then show how to estimate the entropic scale and give some examples.