On the exact region between Chatterjee's rank correlation and Spearman's footrule

Chatterjee's rank correlation $\xi$ has emerged as a popular measure quantifying the strength of directed functional dependence between random variables $X$ and $Y$. If $X$ and $Y$ are continuous, $\xi$ equals Spearman's footrule~$\psi$ for the Markov product of the copula induced by $(X,Y)$ and its transpose. We analyze the relationship between these two measures more in depth by studying the attainable region of possible pairs $(\xi, \psi)$ over all bivariate copulas. In particular, we show that for given $\xi$, the maximal possible value of $\psi$ is uniquely attained by a Fr\'echet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that $\xi\le\psi\le \sqrt{\xi}$ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of $\psi$ for given $\xi$, we give a lower bound based on Jensen's inequality and construct a two-parameter copula family that comes comparably close.