18.24 (2014)

For a group $G$, a function $\phi: G \to \mathbb{R}$ is a quasimorphism if there is a least non-negative number $D(\phi)$ (called the defect) such that $|\phi(gh) - \phi(g) - \phi(h)| \leqslant D(\phi)$ for all $g, h \in G$. A quasimorphism is homogeneous if in addition $\phi(g^n) = n\phi(g)$ for all $g \in G$. Let $\phi$ be a homogeneous quasimorphism of a free group $F$. For any quasimorphism $\psi$ of $F$ (not required to be homogeneous) with $|\phi - \psi| < $ const, we must have $D(\psi) \geqslant D(\phi)/2$. Is it true that there is some $\psi$ with $D(\psi) = D(\phi)/2$?