The exact joint region of Kendall’s tau, Spearman’s footrule and Blomqvist’s beta is fully determined by the already-known pairwise regions of (footrule, beta) and (tau, footrule).

Summary. The manuscript determines the exact attainable set Ω_{τ,φ,β} of joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over all bivariate copulas. It asserts that this set consists precisely of the triples (t,p,b) satisfying the three displayed inequalities −1≤b≤1, (3/16)(1+b)²−1/2≤p≤1−(3/8)(1−b)² and (4/3)p−1/3≤t≤(2/3)p+1/3, which are exactly the known sharp pairwise bounds for (φ,β) and (τ,φ). Consequently, once φ is fixed, β imposes no further sharp restriction on τ. The argument is described as constructive: two one-parameter families of shuffles of M attain the extreme values of τ along the lower (φ,β)-boundary, ordinal sums spread those families through the region, and biaffinity of the concordance function fills the vertical fibres. The paper further claims convexity of Ω, rectangular fixed-footrule sections, an affine symmetry of the fibres about τ=φ, and volume equal to 31/40.

Significance. If the characterization is correct, the work closes a natural gap in the geometry of classical concordance measures by showing that the joint (τ,φ,β)-region is completely determined by the already-known pairwise regions. The structural conclusion—that Blomqvist's beta adds no sharp restriction on Kendall's tau once Spearman's footrule is fixed—is non-obvious and of clear interest to copula theory. The constructive route via shuffles of M, ordinal sums and biaffinity, together with the explicit volume computation, would constitute a solid contribution provided the constructions and boundary verifications hold. Only the abstract is available for this review, so the significance assessment remains conditional on the full proofs.