Gruber P. Convex and Discrete Geometry

96 Convex Bodies
Proposition 6.5. Let C1,...,Cd ∈ C. Then:
(i) V (C1 + t1,...,Cd + td) = V (C1,...,Cd) for all t1,...,td ∈ E d .
(ii) V (mC1,...,mCd) = V (C1,...,Cd) for all rigid motions m : E d → E d .
Remark. More generally, we have the following: let a : E d → E d be an affinity
with determinant det a �= 0. Then
Linearity
V (aC1,...,aCd) =|det a|V (C1,...,Cd).
Mixed volumes are linear in each variable with respect to non-negative linear combinations
of convex bodies.
Proposition 6.6. Let C, D, D2,...,Dd ∈ C. Then
V (λC + µD, D2,...,Dd) = λV (C, D2,...,Dd) + µV (D, D2,...,Dd)
for λ, µ ≥ 0.
Proof. Let λ, µ ≥ 0. The quantities
V � �
λ1(λC + µD) + λ2D2 +···+λd Dd ,
V � �
(λ1λ)C + (λ1µ)D + λ2 D2 +···+λd Dd
have identical polynomial representations in λ1,...,λd. The coefficient of λ1 ···λd
in the first polynomial is
d !V (λC + µD, D2,...,Dd).
The coefficient of λ1 ···λd in the second polynomial can be obtained by representing
the second quantity as a polynomial in λ1λ, λ1µ, λ2,...,λd and then collecting
λ1 ···λd. Thus it is
d !λV (C, D2,...,Dd) + d !µV (D, D2,...,Dd).
Since the coefficients coincide, the proof is complete. ⊓⊔
Continuity
Mixed volume are continuous in their entries.
Theorem 6.8. V (·,...,·) is continuous on C ×···×C.
Proof. The following remark is clear:
Let p, pn, n = 1, 2,..., be homogeneous polynomials of degree d in d
variables such that