Quasi Triangle Inequality for the Lempert function

The (unbounded version of the) Lempert function $l_D$ on a domain $D\subset {\mathbb{C}}^d$ does not usually satisfy the triangle inequality, but on bounded ${\mathcal{C}}^2$-smooth strictly pseudoconvex domains, it satisfies a quasi triangle inequality: $l_D(a,c)\le C(l_D(a,b)+l_D(b,c))$. We show that pseudoconvexity is necessary for this property as soon as $D$ has a ${\mathcal{C}}^1$-smooth boundary. We also give estimates of the Lempert function and of other invariants in some domains which are models for local situations, and derive some general local bounds depending on the regularity of the boundary of a domain.