关于欧氏空间中超平面族的一个猜测

A CONJECTURE ON HYPERPLANES IN EUCLIDEAN SPACE

在欧氏空间R^n放置n个处于一般位置的超平面,由此而分割出的有界凸多面体的集记为{C_1,C_2…,C_r},用v(Ci)表示Ci的顶点个数。本文讨论数v(n,d)=min _i^(max){v(Ci)},这里的最小取遍所有可能的处于一般位置的放置。Kusner指出v(n,2)=4,n≥4,并猜测v(n,d)=2d,n≥2d≥2。本文证明这个猜测在d=3时不正确。

In Euclidean space R^d, let L_n= {A_j} be a family of n hyperplanes in general positions. Let C_1…, C_r be the bounded components of the complement of U A_j; v(C_i) is the number of the convex polytope C_i·Let v(L_n) =max/i {v(C_i)} andv(n, d) =min/L_n {v(L_n)}; minimum ts taken to cover all families L_n of n hyperplanes in general positions in R^d. R. Kusner posed a conjectore that v(n,d)=2d, for n2d2. ln this paper we prore that this conjecture is not true when d=3, and arrive at the following proposition: v(n,3)= 8, n6.