  We consider the closeness testing problem for discrete distributions. The
goal is to distinguish whether two samples are drawn from the same unspecified
distribution, or whether their respective distributions are separated in
$L_1$-norm. In this paper, we focus on adapting the rate to the shape of the
underlying distributions, i.e. we consider \textit{a local minimax setting}. We
provide, to the best of our knowledge, the first local minimax rate for the
separation distance up to logarithmic factors, together with a test that
achieves it. In view of the rate, closeness testing turns out to be
substantially harder than the related one-sample testing problem over a wide
range of cases.
