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

TL;DR

This paper introduces a new similarity-invariant pseudometric on the space of non-degenerate compact convex sets in rom urther explores its topological properties and implications for convex geometry.

Contribution

It defines a novel pseudometric invariant under similarities and characterizes the topology of the quotient space, linking it to the Banach-Mazur compactum and the Hilbert cube.

Findings

The quotient space is homeomorphic to the Banach-Mazur compactum.

The space of convex sets is homeomorphic to the product of the Hilbert cube and urther details.

The pseudometric quantifies differences between convex bodies using classical functionals.

Abstract

In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace of all non-degenerated compact convex sets of $\mathbb R^n$, 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 $\mathcal K^n_*$) is homeomorphic to the Banach-Mazur compactum $BM(n)$, while $\mathcal K^n_*$ is homeomorphic to the topological product $Q\times\mathbb R^{n+1}$, 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.