On the Homothety Conjecture
On the Homothety Conjecture
复制标题
DOI：
10.1512/iumj.2011.60.4299
复制
发表时间：
2009-11
影响因子：
1.1
通讯作者：
E. Werner;Deping Ye
中科院分区：
文献类型：
--
作者：
E. Werner;Deping Ye
设K是Rn中的凸体且> 0.同素异说猜想提出了这样一个问题：K = cK是否意味着K是一个椭球体？这里K是（凸）旋转体，c是仅依赖于的常数。本文证明了在凸体类B n p，1 p 1，l n的单位球类中的位同猜想成立，即证明了（B n p）= c B n当且仅当p = 2.我们还证明了对一般凸体K，若K足够小，则同素性猜想成立。这改善了早期的结果，
Let K be a convex body in R n and > 0. The homothety conjecture asks: Does K = cK imply that K is an ellipsoid? Here K is the (convex) oating body and c is a constant depending on only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies B n p , 1 p 1 , the unit balls of l n ; namely, we show that (B n p ) = cB n if and only if p = 2. We also show that the homothety conjecture is true for a general convex body K if is small enough. This improvs earlier results by