Let C be the set of bivariate copulas, equivalently the set of joint laws of random variables U,V simU(0,1). Define Spearman's rho and Spearman's footrule by rho_S(U,V)=12E[UV]-3 and phi_S(U,V)=1-6E[|U-V|]. Determine exactly the attainable region R=(phi_S(U,V),rho_S(U,V)):(U,V) has a bivariate copula law. Equivalently, for every x∈[-tfrac12,1], determine the extremal values, in particular suprho_S(U,V):phi_S(U,V)=x, and hence the exact upper and lower boundaries of R.

Spearman rho–footrule region problem

Let $\mathcal{C}$ be the set of bivariate copulas, equivalently the set of joint laws of random variables $U,V\sim\mathcal{U}(0,1)$ . Define Spearman's rho and Spearman's footrule by $\rho_S(U,V)=12\mathbb{E}[UV]-3$ and $\phi_S(U,V)=1-6\mathbb{E}[|U-V|]$ . Determine exactly the attainable region $\mathcal{R}=\{(\phi_S(U,V),\rho_S(U,V)):(U,V)\text{ has a bivariate copula law}\}$ . Equivalently, for every $x\in[-\tfrac12,1]$ , determine the extremal values, in particular $\sup\{\rho_S(U,V):\phi_S(U,V)=x\}$ , and hence the exact upper and lower boundaries of $\mathcal{R}$ .

Progress summary

Solved

A new unrefereed preprint claims to finish the long-standing problem, but its result has not yet been independently confirmed.

The problem asks for the full attainable region of Spearman’s rho and footrule for bivariate copulas. Earlier work established the lower boundary and partial upper-bound information; the latest preprint claims a complete characterization.

Known results

Kokol and Stopar (2023) established sharp lower and upper inequalities, with the upper bound known to be sharp only at countably many points.
A 2022 preprint proved the lower boundary for every $\phi(C)\in[-\tfrac12,1]$ and characterized a concave upper estimate attained at some, but not all, points.
A 2024 preprint supplied alternative proofs and improved upper-bound constructions, while explicitly leaving global sharpness open.

August 2026 claimed completion

A preprint dated August 20, 2026 claims that the exact region is determined through a unique optimal coupling and dual potential, thereby resolving the remaining extremal question. The claim is unverified and has not yet been peer reviewed.

Current status (as of August 2026): The lower boundary and substantial partial upper-bound results are established, while a new preprint claims the full region but remains unconfirmed.

Sources

arxiv.org

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Optimal transport and variance formulation
For each prescribed $d\in[0,\tfrac12]$ , among all couplings of $U,V\sim\mathcal{U}(0,1)$ satisfying $\mathbb{E}[|U-V|]=d$ , determine the coupling that minimizes $\operatorname{Var}(|U-V|)$ . This is equivalent to determining the maximal Spearman rho at the prescribed footrule value $\phi_S=1-6d$ .
source: The exact Spearman rho-footrule region via optimal transport with applications to finite rankings, mixability, and Chatterjee's rank correlation

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The exact Spearman rho-footrule region via optimal transport — arXiv

5 hours ago

Progress summary

Equivalent formulations 1

Optimal transport and variance formulation

Primary source

Additional references