Fiber Convex Bodies

Abstract

In this paper we study the fiber bodies, that is the extension of the notion of fiber polytopes for more general convex bodies. After giving an overview of the properties of the fiber bodies, we focus on three particular classes of convex bodies. First we describe the strict convexity of the fiber bodies of the so called puffed polytopes. Then we provide an explicit equation for the support function of the fiber bodies of some smooth convex bodies. Finally we give a formula that allows to compute the fiber bodies of a zonoid with a particular focus on certain zonoids called discotopes. Throughout the paper we illustrate our results with detailed examples.

Gram spectrahedra of ternary quartics

“…In [9] Mathis and Meroni study the fiber body of a compact convex set. Informally, this means the following.…”

“…One may also think about a limit object of Minkowski sums when the number of summands tends to infinity. In [9] the fiber body was then studied in several special cases where the support function of the convex body K is known.…”

“…We start by introducing the fiber body construction from [9]. Let V ⊆ R n+m (m, n ∈ N) be a subspace of dimension n and let W be its orthogonal complement.…”