平移和离散旋转下的泛凸覆盖问题

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry Pub Date : 2023-10-01 DOI:10.1515/advgeom-2023-0021

Mook Kwon Jung, Sang Duk Yoon, Hee-Kap Ahn, Takeshi Tokuyama

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引用次数: 0

摘要

考虑周长为2的平面物体(或等价的长度为2的封闭曲线)允许平移和离散旋转的最小面积通用覆盖。特别地，我们证明了当π允许平移和离散旋转时，解是高为1的等边三角形。我们还给出了长度为2的闭曲线在π /2和2 π /3倍的平移和离散旋转下的凸覆盖。我们证明了对于π /2的倍数的离散旋转，该覆盖的固有闭子集是一个高度小于1的等边三角形的覆盖，并推测该覆盖是面积最小的凸覆盖。最后，我们给出了对于所有整数k =3，在平移和离散旋转为2 π / k的情况下，所有单位段的最小面积凸覆盖。

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Universal convex covering problems under translations and discrete rotations

Abstract We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π /2 and of 2 π /3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π /2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2 π / k for all integers k =3.

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来源期刊

Advances in Geometry 数学-数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.