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 $\tau$, Spearman's rank correlation $\rho$, and Spearman's footrule $\phi$ - relate to Chatterjee's rank correlation $\xi$ when restricted to lower semilinear copulas. First, we provide a complete characterization of the attainable $\tau$-$\rho$ region for this class, thus resolving the conjecture in [18]. Building on this result, we then derive the exact $\tau$-$\phi$ and $\phi$-$\rho$ regions, obtain a closed-form relationship between $\xi$ and $\tau$, and establish the exact $\tau$-$\xi$ region. In particular, we prove that $\xi$ never exceeds $\tau$, $\rho$, or $\phi$. 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.