Characterize the full structural relationship between $L^1$ distribution-function distance and $1$-Wasserstein distance

Characterize the full structural relationship between the $L^1$-distance between multivariate cumulative distribution functions and the $1$-Wasserstein distance beyond the established equality of minimax rates on product Nikol'skii classes.

Background

The paper studies multivariate deconvolution for estimating a signal cumulative distribution function under anisotropic ordinary-smooth noise. In dimensions greater than one, the $1$-Wasserstein distance and the $L^1$-distance between cumulative distribution functions are not generally equivalent. The paper proves that, for product signal measures with anisotropic Nikol'skii marginals, both distances nevertheless have the same minimax rate, using coordinate-wise decoupling of the transport cost.

This result provides only a partial structural connection. The authors explicitly leave unresolved the broader mathematical relationship between the two metrics, including whether their rate agreement reflects a more general principle outside the product-measure setting.