关于凸体覆盖的Hadwiger猜想的两个等价形式

Two equivalent forms of Hadwiger's covering conjecture

令K是一个内部(记作intK)包含原点o的凸体,bdK为其边界,m为覆盖K所需的intK的平移的最小个数.本文证明,存在正实数η和含于η(bdK)的m元点集C1使得C1+int K覆盖K;存在正实数η′、实数γ∈(0,1)和含于η′(bd K)的m元点集C2使得C2+γK覆盖K.基于这两个事实,本文得到关于凸体覆盖的Hadwiger猜想的两个等价形式.本文还引入一个可以替代宗传明提出的攻克Hadwiger猜想的数量方案中的γm(K)的新泛函.

Let K be a convex body containing the origin o in its interior int K, whose boundary is denoted by bd K, and m be the least number of translations of int K needed to cover K. We show that there exist a positive number η and a set C1 of rn points from η（bd K） such that C1 ＋ int K covers K. Similarly, there exist a positive number η＇, a number γ∈ （0, 1）, and a set C2 ofrn points from η＇（bdK） such that C2 ＋γK covers K. These facts yield two equivalent forms of Hadwiger＇s well-known covering conjecture. We also introduce a functional that can take the place of γm（K） in Zong＇s quantitative program to attack Hadwiger＇s covering conjecture.