Revisiting the region determined by Spearman’s ρ and Spearman’s footrule ϕ

Kokol and Stopar (2023) recently studied the exact region ????,?? determined by Spearman’sfootrule ?? and Spearman’s ?? and derived a sharp lower, as well as a non-sharp upper bound for ??given ??. Considering that the proofs for establishing these inequalities are novel and interesting,but technically quite involved we here provide alternative simpler proofs mainly building uponshuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we deriveseveral additional results on shuffle rearrangements and the interplay between diagonal copulasand shuffles which are of independent interest. Moreover we finally show that we can get closerto the (non-sharp) upper bound than established in the literature so far.