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 $Ω(h^{1/3})$ and $O(h^{1/2})$. The upper bound improves to $\varrho(P)\leq O(1+\min\{h^{3/4}Δ,h^{1/2}Δ^{1/2}\})$ if the Euclidean diameter of every hole is most $Δ$ times the Euclidean diameter of $P$; 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：对于平面上带孔的多边形$P$，我们用$\varrho(P)$表示$P$的测地直径与欧几里得直径之比。研究表明，在所有具有$h$个凸孔的凸多边形中，$\varrho(P)$的上确界介于$Ω(h^{1/3})$和$O(h^{1/2})$之间。若每个孔的欧几里得直径至多为$P$的欧几里得直径的$Δ$倍，则上界可改进为$\varrho(P)\leq O(1+\min\{h^{3/4}Δ,h^{1/2}Δ^{1/2}\})$；若每个孔均为\emph{胖}凸多边形，则可改进为$O(1)$。此外，我们证明当$h\rightarrow \infty$时，在具有$h$个凸孔的凸多边形上定义的函数$g(h)=\sup_P \varrho(P)$，与在具有$h$个顶点的几何三角剖分上定义的类似量具有相同的增长率。