  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)).
