Eggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

中文翻译：

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双曲平面和球面上的凸圆盘的周长近似

埃格尔斯顿（逼近平面凸曲线。I. Dowker型定理。PROC。林斯顿。数学。SOC。7 351-377（1957），）证明，在欧几里得平面最好近似凸Ñ边形为凸起圆盘ķ如果我们通过周长偏差来测量距离，则总是用K内接。我们证明了Eggleston陈述的类似物在双曲平面中成立，并给出了一个例子说明它在球体上失效。