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 $h\rightarrow \infty$.

翻译：对于平面上带孔多边形$P$，记$\varrho(P)$为$P$的测地直径与欧几里得直径之比。研究表明，在所有含$h$个凸孔的多边形中，$\varrho(P)$的上确界介于$\Omega(h^{1/3})$与$O(h^{1/2})$之间。若每个孔的直径至多为$\Delta\cdot {\rm diam}_2(P)$，则该上界改进为$O(1+\min\{h^{3/4}\Delta,h^{1/2}\Delta^{1/2}\})$；若每个孔均为“胖”凸多边形，则上界改进为$O(1)$。此外，我们证明当$h\rightarrow \infty$时，所有含$h$个凸孔的多边形上定义的函数$g(h)=\sup_P \varrho(P)$与含$h$个顶点的几何三角剖分上定义的类似量具有相同的增长速率。