Maximal Distortion of Geodesic Diameters in Polygonal Domains

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 $\mathrm{\Omega }(h^{1\mathrm{/}3})$ and $O(h^{1\mathrm{/}2})$. The upper bound improves to $\varrho (P)\le O(1+\min \{h^{3\mathrm{/}4}\mathrm{\Delta },h^{1\mathrm{/}2}{\mathrm{\Delta }}^{1\mathrm{/}2}\})$ if the Euclidean diameter of every hole is most $\mathrm{\Delta }$ 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 $h\to \mathrm{\infty }$.