Aperture-Angle and Hausdorff-Approximation of Convex Figures

TL;DR

This paper proves optimal bounds on aperture angle approximation errors for convex sets and confirms conjectures related to polygon approximation and Hausdorff distance, advancing geometric approximation theory.

Contribution

It establishes the optimal aperture angle bound for inscribed polygons in convex sets and proves a conjecture linking Hausdorff approximation limits to polygon substructures.

Findings

Optimal aperture angle approximation bound of (1 - 2/(k+1)) pi.

Validation of a conjecture relating Hausdorff distance and subpolygon approximation.

Existence of a sub-k-gon within any convex polygon with bounded Hausdorff distance.

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)).