The Spherical Convex Floating Body

TL;DR

This paper introduces the spherical convex floating body, investigates its volume asymptotics, and establishes a new spherical floating area analogous to classical affine surface area.

Contribution

It defines the spherical convex floating body and the floating area, providing foundational properties and establishing its analogy to affine surface area.

Findings

Asymptotic volume difference behavior analyzed

Floating area introduced as a spherical analogue to affine surface area

Several properties of the floating area are established

Abstract

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry. Several properties of the floating area are established.