  For a polygon $P$ with holes in the plane, we denote by $\varrho(P)$ the
ratio between the geodesic and the Euclidean diameters of $P$. It is shown that
over all convex polygons with $h$~convex holes, the supremum of $\varrho(P)$ is
between $\Omega(h^{1/3})$ and $O(h^{1/2})$. The upper bound improves to
$O(1+\min\{h^{3/4}\Delta,h^{1/2}\Delta^{1/2}\})$ if every hole has diameter at
most $\Delta\cdot {\rm diam}_2(P)$; and to $O(1)$ if every hole is a \emph{fat}
convex polygon. Furthermore, we show that the function $g(h)=\sup_P \varrho(P)$
over convex polygons with $h$ convex holes has the same growth rate as an
analogous quantity over geometric triangulations with $h$ vertices when
$h\rightarrow \infty$.
