Level sets of depth measures and central dispersion in abstract spaces

The lens depth of a point have been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points $X_1,X_2$, with the same radius $d(X_1,X_2)$. We study the consistency in Hausdorff and measure distance, of the level sets of the empirical lens depth, based on an iid sample from a general metric space. We also prove that the boundary of the empirical level sets are consistent estimators of their population counterparts. We tackle the problem of how to order random elements in a general metric space by means of the notion of spread out and dispersive order. We present a small simulation study and analyse a real life example.