Fiber Convex Bodies

TL;DR

This paper extends the concept of fiber polytopes to general convex bodies, analyzing properties, explicit support functions, and formulas for specific classes like puffed polytopes and zonoids, with detailed examples.

Contribution

It introduces the fiber body concept for general convex bodies, providing new properties, explicit support functions, and computational formulas for special cases.

Findings

Fiber body of puffed polytopes is strictly convex.

Explicit support function formulas for smooth convex bodies.

Computational method for fiber bodies of zonoids and discotopes.

Abstract

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