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Locating Diametral Points
Results in Mathematics ( IF 1.1 ) Pub Date : 2020-04-01 , DOI: 10.1007/s00025-020-01193-5
Jin-ichi Itoh , Costin Vîlcu , Liping Yuan , Tudor Zamfirescu

Let K be a convex body in $${\mathbb {R}} ^d$$ R d , with $$d = 2,3$$ d = 2 , 3 . We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of $$\mathrm{bd}K$$ bd K , to contain at least one endpoint of a diameter of K . We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E . We also show that such criteria do not exist for $$n\ge 4$$ n ≥ 4 points.

中文翻译:

定位直径点

令 K 为 $${\mathbb {R}} ^d$$ R d 中的凸体,其中 $$d = 2,3$$ d = 2 , 3 。我们为由 $$\mathrm{bd}K$$ bd K 的 1、2 或 3 个点组成的集合 E 确定了尖锐的充分条件,以包含至少一个直径为 K 的端点。我们还将其扩展到具有内在度量的凸面。我们的条件是 E 中点的完整角度之和的上限。我们还表明,对于 $$n\ge 4$$ n ≥ 4 分,这样的标准不存在。
更新日期:2020-04-01
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