---
title: Aperture-Angle and Hausdorff-Approximation of Convex Figures
url: https://www.emergentmind.com/papers/0702090
type: paper
arxiv_id: '0702090'
arxiv_url: https://arxiv.org/abs/0702090
published: '2007-02-16'
categories:
- cs.CG
- math.MG
---

# Aperture-Angle and Hausdorff-Approximation of Convex Figures

## Abstract

The aperture angle alpha(x, Q) of a point x not in 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 Q of C is the minimum aperture angle of any x in C Q 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 Q of C with aperture angle approximation error (1 - 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 sigma, 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 1/(k+1) * sin(pi/(k+1)).