A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of R-n

作者：Merino Bernardo Gonzalez; Jonard Perez Natalia^*

来源：Topology and Its Applications, 2015, 194: 125-143.

DOI：10.1016/j.topol.2015.08.002

• Hyperspace of convex sets
• Convex body
• Banach-Mazur distance
• Hausdorff metric
• Group of similarities
• Inner and outer radii
• Geometric inequalities

In this work we define a new pseudometric in K-*(n), the hyperspace of all non-degenerated compact convex sets of]R-n, which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of similarities on K-*(n)) is homeomorphic to the Banach-Mazur compactum BM(n), while K-*(n) is homeomorphic to the topological product Q X Rn+1, where Q stands for the Hilbert cube. Finally we will show some consequences in convex geometry, namely, we measure how much two convex bodies differ (by means of our new pseudometric) in terms of some classical functionals.

• 出版日期2015-10