Approximation of zonoids by zonotopes in fixed directions (Q1329189)

The authors give a constructive proof for the following result: If \(Z\) is a zonoid (a ``continuous sum'' of segments) in \(\mathbb{R}^ d\) and if \(u_ 1,\dots,u_ k\) are unit vectors, there always exists a zonotope \(Z'\) which is the sum of at most \(k\) segments and such that \(Z\) and \(Z'\) have the same supporting hyperplanes with outer normals \(\pm u_ 1,\dots, \pm u_ k\). In the authors' approach, \(Z'\) is obtained as the solution of a linear program. As an application to stochastic geometry, the authors describe a method for the estimation of the directional distribution of a random fibre process. Also some further geometric consequences of the main result are described; they are related to the interpretation of zonoids as projection bodies.