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：多边形域中测地线直径的最大失真 翻译后的摘要： 对于一个带洞的平面多边形$P$，我们用$\varrho(P)$表示其测地线直径和欧几里得直径之比。本文证明，在所有凸带$h$个凸洞的多边形中，$\varrho(P)$的最大值在$\Omega(h^{1/3})$和$O(h^{1/2})$之间。如果每个洞的直径不超过$\Delta\cdot {\rm diam}_2(P)$，则上边界可改进为$O(1+\min\{h^{3/4}\Delta,h^{1/2}\Delta^{1/2}\})$；如果每个洞都是\emph{fat}凸多边形，则上界为$O(1)$。此外，我们还证明，在$h\rightarrow \infty$时，凸带$h$个顶点的几何三角剖分和凸多边形中函数$g(h)=\sup_P \varrho(P)$的增长速率是相同的。