  In this paper we study the global geometry of the Kobayashi metric on
"convex" sets. We provide new examples of non-Gromov hyperbolic domains in
$\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex,
bounded and unbounded. Our first aim is to prove that if $\Omega$ is a bounded
weakly linearly convex domain in $\mathbb{C}^n,\,n\geq 2,$ and $S$ is an affine
complex hyperplane intersecting $\Omega,$ then the domain $\Omega\setminus S$
endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next
we localize this result on Kobayashi hyperbolic convex domains. Namely, we show
that Gromov hyperbolicity of every open set of the form $\Omega\setminus S',$
where $S'$ is relatively closed in $\Omega$ and $\Omega$ is a convex domain,
depends only on that how $S'$ looks near the boundary, i.e., whether $S'$ near
$\partial\Omega$ (Theorem 1.7). We close the paper with a general remark on
Hartogs type domains. The paper extends in an essential way results in [6].
