A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of \mathbb Rⁿ

2015/03/07 by Bernardo González Merino, Merino, Bernardo González, Natalia Jonard-Pérez +1
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.GT #math.MG

paper · pdf · doi:10.48550/arxiv.1503.02184

openalex publication_date 2015/03/07 · arxiv created 2015/05/09 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we define a new pseudometric in \mathcal Kⁿ_*, the hyperspace of all non-degenerated compact convex sets of \mathbb Rⁿ, 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 \mathcal Kⁿ_*) is homeomorphic to the Banach-Mazur compactum BM(n), while \mathcal Kⁿ_* is homeomorphic to the topological product Q×\mathbb 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.

Citations

• The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

Related

• Hyperspaces of convex bodies of constant width
• Hyperspaces of smooth convex bodies up to congruence
• On convex bodies of constant width
• The hyperspace of k-dimensional closed convex sets
• On k-convex hulls