特殊凸体边界小位似体覆盖

Covering the Boundary of Special Convex Bodies with Smaller Homothetic Copies

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R^n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。

For Hadwiger conjecture that the least number c(K) of translates of the interior of K needed to cover K is at most 2 n , according to the fact that c(K) equals the least number of smaller homothetic copies of K with the same homothetic ratio needed to cover the boundary bd K of K , drawing on the continuous functional to estimate the value of c(K), studying the problem of estimating the value of the smallest homothetic ratio γ m(K) that bd K can be covered by m translates of γ m(K)K . The value of γ 4(K) and γ 8(K) when K is a regular tetrahedron, as well as the value of γ 6(K),γ 7(K) and γ 8(K) are obtained when K is a regular octahedron are obtained. It is proved that γ 2 n (K)=Γ 2 n (K)=Γ 2 n-1 (D) when K is a cylinder in R^n with the n -1 dimensional convex set D as the base, where Γ m(K) is the smallest value of the homothetic radio γ that K can be covered by m translates of γK .