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Separation bodies: a conceptual dual to floating bodies

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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.

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  1. Mathematisches Institut, Albert-Ludwigs-Universität, D-79104, Freiburg i. Br., Germany

    Rolf Schneider

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  1. Rolf Schneider
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Correspondence to Rolf Schneider.

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Communicated by Adrian Constantin.

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Schneider, R. Separation bodies: a conceptual dual to floating bodies. Monatsh Math 193, 157–170 (2020). https://doi.org/10.1007/s00605-020-01443-2

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