On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas

We explore how the classical concordance measures - Kendall's $τ$, Spearman's rank correlation $ρ$, and Spearman's footrule $ϕ$ - relate to Chatterjee's rank correlation $ξ$ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable $τ$-$ρ$ region for this class, thus resolving the conjecture in [18]. Building on this result, we then derive the exact $τ$-$ϕ$ and $ϕ$-$ρ$ regions, obtain a closed-form relationship between $ξ$ and $τ$, and establish the exact $τ$-$ξ$ region. In particular, we prove that $ξ$ never exceeds $τ$, $ρ$, or $ϕ$. Our results clarify the relationship between undirected and directed dependence measures and reveal novel insights into the dependence structures that result from lower semilinear copulas.