多边形域中测地直径的最大扭曲
Maximal Distortion of Geodesic Diameters in Polygonal Domains
AI总结：
研究了多边形域中测地直径与欧几里得直径的比值，证明了在具有凸孔的凸多边形中，该比值的上界随孔数变化而变化，并探讨了不同条件下该比值的性质。
摘要：
对于平面中带有孔的多边形P，我们用ρ(P)表示测地直径与欧几里得直径的比值。证明了在具有h个凸孔的凸多边形中，ρ(P)的上确界在Ω(h^{1/3})和O(h^{1/2})之间。如果每个孔的欧几里得直径最多是P的欧几里得直径的Δ倍，则上界改进为ρ(P)≤O(1+min{h^{3/4}Δ,h^{1/2}Δ^{1/2}})；如果每个孔都是肥的凸多边形，则上界为O(1)。此外，我们证明当h趋近于无穷大时，凸多边形具有h个凸孔时的函数g(h)=sup_P ρ(P)的增长率与几何三角剖分具有h个顶点时的相应量的增长率相同。
英文摘要：
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 $h\rightarrow \infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
