Separation bodies: a conceptual dual to floating bodies

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.