  The lens depth of a point has 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 and Y, with the same radius d(X, Y). We study the consistency in
Hausdorff and measure distance, of the level sets of the empirical lens depth,
based on an iid sample on a general metric space. We also prove that the
boundary of the empirical level sets are consistent estimators of their
population counterparts, and analyze two real-life examples
