Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies

JI Dong-hai; Lv De-jing; MA Ze-min

doi:10.15938/j.jhust.2019.02.017

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Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies

更新时间：2026-08-31

- Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies
- Vol. 24, Issue 2, Pages: 115-120(2019)
- 作者机构：
  
  哈尔滨理工大学理学院,哈尔滨,150080
- 作者简介：
- 基金信息：
- DOI：10.15938/j.jhust.2019.02.017
  CLC： O177
- Published：2019
- 稿件说明：
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JI Dong-hai, Lv De-jing, MA Ze-min. Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies[J]. 2019, 24(2): 115-120.
DOI：

JI Dong-hai, Lv De-jing, MA Ze-min. Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies[J]. 2019, 24(2): 115-120. DOI： 10.15938/j.jhust.2019.02.017.

摘要

针对覆盖n维凸体K所需K的内部的平移的最小数目c(K)不超过2n的Hadwiger猜想，依据c(K)等于覆盖K的边界bdK所需K的位似系数相同的小位似体的最小数目这一核心结果，借鉴将c(K)的估值问题连续化的方法，研究了用m个K的位似系数相同的小位似体覆盖bdK所需最小位似系数γm(K)精确值的估算问题。得到了当K是正四面体时γ4(K)和γ8(K)的值以及当K是正八面体时γ6(K)，γ7(K)和γ8(K)的值，并证明了当K是Rn中以n－1维凸体D为底的柱体时γ2n(K)=Γ2n(K)=Γ2n－1(D)，其中Γm(K)表示用m个K的位似系数相同的小位似体覆盖K所需最小位似系数。

Abstract

For Hadwiger conjecture that the least number c(K) of translatesof the interior of K needed to cover K is at most 2n

according to the factthat c(K) equals the least number of smaller homothetic copies of K with thesame homothetic ratio needed to cover the boundary bdK of K

drawing on thecontinuous functional to estimate the value of c(K)

studying the problem ofestimating the value of the smallest homothetic ratio γm(K) that bdK can becovered by m translates of γm(K)K. The value of γ4(K) and γ8(K) when K is aregular tetrahedron

as well as the value of γ6(K)

γ7(K) and γ8(K) areobtained when K is a regular octahedron are obtained. It is proved that γ2n(K)=Γ2n(K)=Γ2n－1(D)when K is a cylinder in Rn with the n－1 dimensional convex set Das the base

where Γm(K) is the smallest value of the homothetic radio γ that Kcan be covered by m translates of γK..

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