R^n空间中单位球面覆盖的半径问题

Radius of A Minimal Ball-covering of R^n Spaces

Banach空间X中的一个闭球族B是X的球覆盖,如果B中的任一元素不包含原点作为其内点,且B中元素之并覆盖了X的单位球面SX.一个球覆盖B称为是极小的当且仅当B的势小于或等于X中所有球覆盖的势.文献[1]证明了在Rn中球覆盖的极小势为n+1,本文重点利用文献[4]所给出的n维空间中n-单形与其外接超球面间的若干关系,证明了在有限维欧氏空间Rn中极小球覆盖的最小半径为n/2,且当极小球覆盖中(n+1)个球的球心恰好为球面n/2SX的内接正则n-单形的顶点时可以取到.

We say A family B of closed balls in a Banach space X is a ball-covering of X if every ball in B does not contain the origin in its interior and whose union covers the unit sphere Sx of X, and a ball-covering B is said to be minimal if the cardinal of B is less than or equal to the cardinal of every ball-covering of X. Article [1] showed that R^n admits a minimal ball-covering of n＋1 balls. This article then presents that for n≥2, the n smallest radius of all minimal ball-coverings of R^n is n/2 and it is attained whenever the centers of the n＋1 balls n of a minimal ball-eovering are the vertices of a regular inscribed n-simplex of the sphere n/2 Sx.