Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^1 d(f(t),w)\,dt$ for every $w\in X$. If such $m_f\in X$ exists then we call $f$ as $D$-integrable with $D$-integral $m_f$. ($I=[0,1]$.)
Question: Given a finitely generated group $G$ (with a usual word distance), does there exist a condition on the set of $D$-integrable functions $f: I\to G$ (which uses $D$-integrability) for the Gromov $\delta$-hyperbolicity of $G$? I.e, what is the characterization of hyperbolicity in terms of $D$-integrability?
(It seems to me that the real question here should be: how can one extract group theoretic intel from $D$-integrability?)