与Lebesgue测度有关的紧凸集的超空间的拓扑结构

The Topological Structures of Hyperspaces of Compact Convex Sets Concerned with Lebesgue Measure

本文主要证明了欧氏平面上,面积不超过某给定正数的紧凸集全体,赋予Hausdorff度量拓扑构成的超空间,是一个AR;还证明了[0,1]×[0,1]中,Lebesgue测度不超过某正数m_0(m_0<1)的紧凸集全体同胚于Hilbert方体Q=[-1,1]~ω.

In this paper,we mainly proved that the hyperspace of all compact convex sets which not exceeding a given positive constant,endowed with the Hausdorff metric topology,is homeomorphic to an AR; And also proved that the hyperspace of all compact convex sets which Lebesgue measure not exceeding m_0( m_0<1) in[0,1]×[0,1],is homeomorphic to the Hilbert cube Q =[-1,1]~ω.