Weighted floating bodies and polytopal approximation

TL;DR

This paper develops asymptotic results for weighted floating bodies and applies them to prove the existence of floating areas across various geometries, including sphere, hyperbolic space, and Hilbert geometries, advancing approximation theories.

Contribution

It introduces new asymptotic results for weighted floating bodies and offers novel proofs for floating areas in diverse geometric settings, linking approximation methods with geometric analysis.

Findings

Existence of floating areas on the sphere, hyperbolic space, and Hilbert geometries.

New asymptotic approximation results in these geometries.

Unified approach connecting floating bodies with approximation theory.

Abstract

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floating areas are combined to derive new asymptotic approximation results on the sphere, in hyperbolic space and in Hilbert geometries.