The floating body in real space forms

TL;DR

This paper introduces a unified approach to studying floating bodies across different space forms, extending classical Euclidean results to spherical and hyperbolic geometries, and linking volume derivatives to a new surface measure.

Contribution

It develops a unifying framework for floating bodies in various space forms and establishes a relation between volume derivatives and a new surface area measure called floating area.

Findings

In Euclidean space, floating area equals affine surface area.

Established a relation between volume derivative and floating area in space forms.

Extended floating body theory to hyperbolic space.

Abstract

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.