Zonotopes With Large 2D Cuts
Thilo R\"orig, Nikolaus Witte, and G\"unter M. Ziegler
TL;DR
This paper investigates the maximum number of vertices in 2D sections of high-dimensional zonotopes, establishing asymptotic optimal bounds for all dimensions greater than or equal to two.
Contribution
It extends known results for 3D zonotopes to all higher dimensions, providing asymptotically optimal bounds for the number of vertices in 2D cuts.
Findings
For d-dimensional zonotopes with n zones, 2D sections can have (n^{d-1}) vertices.
The bounds are asymptotically optimal for all fixed d.
Generalizes previous 3D results to higher dimensions.
Abstract
There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs'' by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.
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Taxonomy
TopicsMorphological variations and asymmetry · Computational Geometry and Mesh Generation · Mathematical Dynamics and Fractals