Gruber P. Convex and Discrete Geometry

202 Convex Bodies
Outer Billiards
Fig. 10.3. Outer billiard
Let B be a planar, strictly convex body (see Fig. 10.3). Given x0 ∈ E 2 \ B, consider
the support line of B through x0 such that B is on the left side of this line if viewed
from x0. Letx1 be the mirror image of x0 with respect to the touching point. The
dynamical system x0 → x1 is called the outer billiard determined by B. In our
context outer billiards are important for the approximation of planar convex bodies,
see Sect. 11.2 and the article of Tabachnikov [984] cited there. The latter proved a
result on outer billiards which yields an asymptotic series development for best area
approximation of a planar convex body by circumscribed polygons as the number of
edges tends to infinity.
11 Approximation of Convex Bodies and Its Applications
Most approximation results in convex geometry belong to one of the following types:
Approximation by special convex bodies, such as ellipsoids, simplices, boxes,
or by special classes of convex bodies, for example centrally symmetric convex
bodies, analytic convex bodies, or zonotopes
Asymptotic best approximation by convex polytopes with n vertices or facets as
n →∞
Approximation of convex bodies by random polytopes, i.e. the convex hull of n
random points
Asymptotic approximation by random polytopes as n →∞
There are many sporadic results of the first type scattered throughout the convexity
literature. Pertinent results of a more systematic nature can be found in the context
of the maximum and minimum ellipsoids in the local theory of normed spaces.
The first asymptotic formulae for best approximation of a convex body with
respect to a metric were given by L. Fejes Tóth [329] for d = 2 in the early 1950s.
Asymptotic formulae for general d were first proved by Schneider [905] for the