低维Busemann-Petty测度问题的同伦形式

An Isomorphic Version of the LowerDimensional Busemann-PettyProblems for Measures

摘要: 低维Busemann-Petty测度问题是指: 对其有适当密度函数的Borel测度µ及n-维欧氏空间的两个中心对称凸体K和L来说，若它们被任意的n-i-维子空间所截，所得的n-i–维截面体的测度满足µ(K∩ξ^⊥)≤µ(L∩ξ^⊥)，其中ξ∈G(n,i)(1≤i ≤n)，那么其n-维凸体K和L的测度µ(K)≤µ(L)是否成立？在己有文献中，获得了与Rubin及Zhang关于低维Busemann-Petty问题的n-维体积形式相一致的结论。本文证明了这个结论的同伦形式，即在上述条件下，对任意的1≤i≤n，有µ(K)≤n^i/2µ(L)成立。

Abstract: The lower dimensional Busemann-Petty (LDBP) problem for arbitrary measures asks: For a given Borel measure µ with appropriate density and two origin-symmetric convex bodies K and L, does the assumption that µ(K∩ξ^⊥)≤µ(L∩ξ^⊥) holds for any ξ∈G(n,i)(1≤i ≤n) imply that µ(K)≤µ(L)? It was proved that the problem has the same answer as Rubin and Zhang's solutions to the LDBP problem for volumes. In this paper we show an isomorphic version of this result. Namely, if the above conditions hold, then µ(K)≤n^i/2µ(L) for any 1≤i≤n.