Discrete & Computational Geometry

, Volume 40, Issue 3, pp 414–429

# Aperture-Angle and Hausdorff-Approximation of Convex Figures

• Hee-Kap Ahn
• Sang Won Bae
• Otfried Cheong
• Joachim Gudmundsson
Article

## Abstract

The aperture angle α(x,Q) of a point x Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon QC is the minimum aperture angle of any xCQ with respect to Q. We show that for any compact convex set C in the plane and any k>2, there is an inscribed convex k-gon QC with aperture angle approximation error $$(1-\frac{2}{k+1})\pi$$ . This bound is optimal, and settles a conjecture by Fekete from the early 1990s.

The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance σ, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k+1)-gon. This implies the following result: For any k>2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most  $$\frac{1}{k+1}\sin\frac{\pi}{k+1}$$ .

## Keywords

Hausdorff approximation Aperture angle Convex figure Subpolygon

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## Authors and Affiliations

• Hee-Kap Ahn
• 1
• Sang Won Bae
• 2
• Otfried Cheong
• 2
• Joachim Gudmundsson
• 3
1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
2. 2.Division of Computer ScienceKorea Advanced Institute of Science and TechnologyDaejonKorea
3. 3.National ICT Australia LtdSydneyAustralia