Floating functions

TL;DR

This paper introduces floating bodies for convex sets and extends the concept to floating functions and measures, establishing their asymptotic behavior and defining a new affine invariant similar to Euclidean affine surface area.

Contribution

It generalizes floating bodies to convex sets, functions, and measures, and introduces a new affine invariant with properties akin to Euclidean affine surface area.

Findings

Established asymptotic behavior of integral differences for log concave functions

Defined floating functions for convex and log concave functions and measures

Introduced a new affine invariant similar to Euclidean affine surface area

Abstract

We introduce floating bodies for convex, not necessarily bounded subsets of $\mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of the integral difference of a log concave function and its floating function. This gives rise to a new affine invariant which bears striking similarities to the Euclidean affine surface area.