Gruber P. Convex and Discrete Geometry

210 Convex Bodies
Theorem 11.4. Let C ∈ Cp be of class C2 with Gauss curvature κC > 0. Then there
is a constant δ = δ2,d−1 > 0, depending only on d, such that
(1) δ V (C, P c ⎛
�
δ
(n) ) ∼ ⎝ κC(x)
2
1
⎞ d+1
d−1
d+1 dσ(x) ⎠
1
as n →∞.
bd C
n 2
d−1
Here, σ is the ordinary surface area measure on bd C. The integral in the asymptotic
formula (1) is the affine surface area of C, a notion from affine differential geometry.
It is invariant with respect to volume preserving affinities of the convex body C.The
constant δ = δ2,d−1 was introduced in [427], where we wrote div (for Dirichlet
and Voronoĭ) instead of δ. It is related to Dirichlet–Voronoĭ tilings, see Sect. 32.1.
We have,
δ2,2 = 5
18 √ 3 , δ2,d−1∼ d
as d →∞.
2πe
For the value of δ2,2, see Fejes Tóth [329] and the author [425, 427]. For a proof of
the asymptotic formula for δ2,d−1, see Proposition 33.1.
Proof. Let λ>1. We start with some preparations. For p ∈ bd C, letH be the
unique support hyperplane of C at p. Choose a Cartesian coordinate system in H
with origin at p. Together with the interior unit normal vector of bd C at p, itforms
a Cartesian coordinate system in E d . The lower part of bd C with respect to the last
coordinate then can be represented in the form:
� s, f (s) � : s ∈ C ′ ,
where “ ′ ” denotes the orthogonal projection of Ed onto H and f is a convex function
on C ′ such that f | relint C ′ is of class C2 .Foru∈ relint C ′ , define the quadratic form
qu = qpu by
qu(s) = �
fxi ,x j (u)sisj for s = (s1,...,sd−1) ∈ H.
i, j
Let qp = qpp. The Gauss curvature κC(u) at the point x = � u, f (u) � ∈ bd C,
u ∈ relint C ′ , is then given by
κC(u) =
det qu
� � � . d+1 2�
1 + grad f (u) 2
We also write κC(x) for κC(u). Since κC > 0, the quadratic forms qu are all positive
definite. Since f is of class C 2 , their coefficients are continuous. Hence we may
choose an open convex neighbourhood U ′ ⊆ C ′ of p in H such that
1
λ qp(s) ≤ qu(s) ≤ λqp(s) for s ∈ H, u ∈ U ′
1
λ det qp ≤ det qu ≤ λ det qp for u ∈ U ′
1
λ κC(u) ≤ det qp ≤ λκC(u) for u ∈ U ′