Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 159
Let P be a convex polytope with V (P) = V (K ) such that all its facets are among
the faces
Gi = � x : ui · x = h P(ui) � ∩ P.
For the proof of the theorem we have to show the following:
(7) The total free surface energy of P is greater than that of K ,
unless P is a translate of K .
To see this, we first show that:
(8) V � (1 − λ)P + λK � is a positive polynomial in λ for 0 ≤ λ ≤ 1, and
V � (1 − λ)P + λK � = (1 − λ) d V (P)
+ (1 − λ) d−1λ � m�
εi v(Gi) +
n�
δi εi v(Gi) �
+ O(λ 2 ) as λ → 0.
i=1
G i facet of P
i=m+1
G i facet of P
The first statement follows from Minkowski’s (see Fig. 8.3) theorem 6.5 on mixed
volumes. To see the second, note that (1 − λ)P + λK can be dissected into the polytope
(1 − λ)P, cylinders with basis (1 − λ)Gi and height λεi if i ≤ m, respectively,
λδiεi if i > m, where Gi is a facet of P, and polytopes of total volume O(λ 2 ).
Second, since V (P) = V (K ), the theorem of Brunn–Minkowski implies that
(9) The function f (λ) = V � (1 − λ)P + λK � 1 d − (1 − λ)V (P) 1 d − λV (K ) 1 d
for 0 ≤ λ ≤ 1 with f (0) = f (1) = 0 is strictly concave, unless P is a
translate of K , in which case it is identically 0.
Note that V (P) = V (K ). Since, by (8), f is differentiable for 0 ≤ λ ≤ 1, Proposition
(9) implies that
(10) f ′ (0) ≥ 0, where equality holds if and only if P is a translate of K .
o K
(1 − λ)P
Fig. 8.3. Proof of Wulff’s theorem
(1 − λ)P + λK