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.