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We provide a characterization of the radii minimal projections of polytopes onto j-dimensional subspaces in Euclidean space E. Applied on simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part… (More)

- René Brandenberg
- 2004

In der Folge dieser Resultate wurden die vier grundlegenden Radien (Dicke, Durchmesser, In-und Umkugelradius) in einer Vielzahl von Arbeiten behandelt. In den Literaturverzeichnissen findet man u.a. Artikel zu speziellen geometrischen oder Verallgemeinerungen von Jung und Steinhagens Theoremen für allgemeine Minkowski-Räume [9, 57, 26]. Wie bereits erwähnt,… (More)

- René Brandenberg, Thorsten Theobald
- Applicable Algebra in Engineering, Communication…
- 2003

We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space n . Explicitly, the computation of a smallest enclosing cylinder in 3 is reduced to the computation of a smallest circumscribing cylinder. We improve existing polynomial formulations to compute the locally extreme… (More)

- René Brandenberg, Stefan König
- Symposium on Computational Geometry
- 2011

This paper deals with the containment problem under homothetics, a generalization of the minimal enclosing ball (MEB) problem. We present some new geometric identities and inequalities in the line of Jung's Theorem and show how those effect the hope on fast approximation algorithms using small core-sets as they were developed in recent years for the MEB… (More)

We review some recent results on inner and outer j-radii of simplices and general convex bodies. In particular, we discuss two lines of research whose present-day developments were strongly influenced by work of Bernulf Weißbach: radii of regular simplices and geometric inequalities among the radii. MSC 2000: 52A20, 52A40, 52B12

- René Brandenberg
- Discrete & Computational Geometry
- 2005

There are three types of regular polytopes which exist in every dimension d: regular simplices, (hyper-) cubes, and regular cross-polytopes. In this paper we investigate two pairs of inner and outer j-radii (rj, Rj) and (r̄j, R̄j) of these polytopes (inner and outer radii classes are almost always considered in pairs, such that for a 0-symmetric body K and… (More)

- René Brandenberg, Lucia Roth
- COCOA
- 2008

- René Brandenberg, Lucia Roth
- Comp. Opt. and Appl.
- 2011

In this paper, we show that for any dimension d ≥ 3 there exists a body of constant breadth C, such that its projection onto any 2-plane is non-spherical. We call such a body totally non-spherical. The circumradius of the projection of any totally non-spherical body C of constant breadth onto any 2-plane is bigger than the half diameter of C. Showing the… (More)

- René Brandenberg, Abhi Dattasharma, Peter Gritzmann, David G. Larman
- Discrete & Computational Geometry
- 2004

In this paper, we show that for any dimension d ≥ 2 there exists a non-spherical strongly isoradial body i.e., a non-spherical body of constant breadth, such that its orthogonal projections on any subspace has constant inand circumradius. Besides the curiosity aspect of these bodies, they are highly relevant for the analysis of geometric inequalities… (More)