2013/12/15 by Sergey Antonyan, Antonyan, Sergey, Natalia Jonard-Pérez +3
Mathematics · #52A20 #52A99 #54B20 #54C55 #57N20 #57S10 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #math.GT #math.MG #msc:52A20 #msc:52A99 #msc:54B20 #msc:54C55 #msc:57N20 #msc:57S10
paper · pdf · doi:10.48550/arxiv.1312.4141
12 pages
arxiv created 2013/12/15 · arxiv updated 2013/12/17
Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cwD(Rn) of all compact convex subsets with constant width d∈ D, where D is a convex subset of [0,∞). Our main result states that cwD(Rn) is homeomorphic to D× Rn× Q, where Q denotes the Hilbert cube. We also prove that the hyperspace crwD(Rn), consisting of all pairs of compact convex sets of constant relative width d∈ D is homeomorphic to cwD(Rn). In particular, we prove that the hyperspace cw(Rn) of all compact convex bodies of constant width, as well as the hyperspace crw(Rn) of all pairs of compact convex sets of constant relative positive width are homeomorphic to Rn+1× Q.