Gruber P. Convex and Discrete Geometry

11 Approximation of Convex Bodies and Its Applications 217
Remark. For refinements and information on the form of Pn, see the author’s
articles [439] (d = 3) and [443] (general d). The results in these papers deal with
normed spaces and make use of the generalized surface area as treated in Sect. 8.3.
One of the tools used there is a result of Diskant [274] which extends Lindelöf’s
theorem to normed spaces.
Heuristic Observations
The asymptotic formula for best approximation of C by inscribed convex polytopes
is as follows, see Gruber [427].
δ V (C, P i γ
n ) ∼
2
A(C) d+1
d−1
1
n 2
d−1
as n →∞,
where γ = γ2,d−1 > 0 is a suitable constant depending on d, and
�
A(C) = κC(x) 1
d+1 dσ(x)
bd C
the affine surface area of C. A result of Bárány [67] and Schütt [920] on random
polytopes states the following, where E � δ V (C, Qk) � stands for the expectation of
the difference of the volume of C and the volume of the convex hull Qk of k random
points uniformly distributed in C,
E � δ V (C, Qk) � ∼ cd A(C) 1
k 2
d+1
as k →∞.
Here cd > 0 is a suitable constant depending on d. These two asymptotic formulae
seem to say that random approximation is less efficient than best approximation (put
k = n), but, as observed by Bárány, this is the wrong comparison to make. Being
the convex hull of k random points in C, the random polytope Qk in general has less
than k vertices. Actually, for the expectation E � v(Qk) � of the number of vertices of
Qk, wehave
Denote this expectation by n. Then,
E � v(Qk) � ∼ cd A(C) k d−1
d+1 as k →∞.
E � δ V (C, Qk) � ∼ c d+1
d−1
d
d+1
A(C) d−1
1
n 2
d−1
as n →∞.
For d = 2, 3Bárány [68] proved an even stronger result. Since Mankiewicz and
Schütt [686] showed that
c d+1
d−1
d
1
2γ2,d−1 → 1asd →∞,
we see that for large d, random approximation is almost as good as best approximation.
This is an example of the following vague principle.