Gruber P. Convex and Discrete Geometry

160 Convex Bodies
Third, by (8) and (9),
f ′ (λ) = 1
d V � (1 − λ)P + λK � 1 d −1� − d(1 − λ) d−1 V (P)
+ � − (d − 1)(1 − λ) d−2 λ + (1 − λ) d−1�� � + � � + O(λ) �
+ V (P) 1 d − V (K ) 1 d as λ → 0.
Since V (P) = V (K ), we thus have,
(11) f ′ (0) = 1
d V (P) 1 d −1� − dV(P) + � + � �
≥ 0
by (10) or, equivalently,
(12)
m�
εiv(Gi) +
n�
δi εi v(Gi)
Therefore
(13)
i=1
G i facet of P
n�
i=1
G i facet of P
i=m+1
G i facet of P
≥ dV(P) = dV(K ) = m�
εi v(Gi) ≥
m�
i=1
εi v(Fi).
εi v(Fi).
i=1
If there is equality in (13), there must be equality in (12) and thus in (11). Then (10)
shows that P is a translate of K . This concludes the proof of (7) and thus of the
theorem. ⊓⊔
Remark. The evolution of curves in the plane or on other surfaces has attracted
much interest in recent years. An example is the curvature flow where the velocity
of a point of the curve is proportional to the curvature at this point and the direction
is orthogonal to the curve. See Sect. 10.3. In recent years, Wulff’s theorem gave rise
to several pertinent results. See, e.g. the articles of Almgren and Taylor [24] and
Yazaki [1032].
Packing of Balls and Wulff Polytopes
A problem due to Wills [1027] where Wulff’s theorem plays a role is the following:
Consider a lattice L which provides a packing of the unit ball B d .LetP be
a convex polytope, with vertices in L, and for n ∈ N consider the finite packings
{B d + l : l ∈ L ∩ nP}. It turns out that the so-called parametric density δρ(B d ,
L ∩ nP) of these packings approaches the density δ(B d , L) = V (B d )/d(L) of
the lattice packing {B d + l : l ∈ L} rapidly as n → ∞ if P is close to a certain
Wulff polytope. Tools for the proof are Minkowski’s theorem on mixed volumes
and Ehrhart’s polynomiality theorem for lattice polytopes. For more information and
references, see Böröczky [155], in particular Sect. 13.