Separation bodies: a conceptual dual to floating bodies

TL;DR

This paper introduces separation bodies as a dual concept to floating bodies in convex geometry, exploring their properties and potential applications in the study of random polytopes generated by intersections of random halfspaces.

Contribution

It defines separation bodies in a measure-theoretic framework, establishing their foundational properties and illustrating their role as conceptual duals to floating bodies.

Findings

Separation bodies are well-defined convex sets related to hyperplane measures.

They serve as a dual tool to floating bodies in convex geometry.

Initial examples suggest their usefulness in analyzing random polytopes.

Abstract

Let $K$ be a convex body in Euclidean space ${\mathbb R}^d$, and let a translation invariant, locally finite Borel measure on the space of hyperplanes in ${\mathbb R}^d$ be given. For $\delta\ge 0$, we consider the set of all points $x$ for which the set of hyperplanes separating $K$ and $x$ has measure at most $\delta$. This defines the separation body of $K$, with respect to the given measure and the parameter $\delta$. Separation bodies are meant as conceptual duals to floating bodies, and they are expected to play a role in the investigation of random polytopes generated as intersections of random halfspaces, in a similar way that floating bodies are useful for studying convex hulls of random points. After discussing some elementary properties of separation bodies, we carry out first examples to this effect.