Flag numbers and floating bodies

TL;DR

This paper explores weighted floating bodies of polytopes, revealing their dependence on complete flags and introducing flag simplices, with applications to spherical and hyperbolic geometries and contrasting behaviors with Euclidean space.

Contribution

It introduces flag simplices to connect metric and combinatorial structures, and extends floating body analysis to spherical and hyperbolic spaces with new asymptotic results.

Findings

Weighted volume depends on complete flags of polytopes

Flag simplices bridge metric and combinatorial structures

Floating bodies in spherical and hyperbolic spaces can behave differently from Euclidean cases

Abstract

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.