Mixed volume

Definition

Let K 1 , K 2 , … , K r be convex bodies in R n and consider the function

f ( λ 1 , … , λ r ) = V o l n ( λ 1 K 1 + ⋯ + λ r K r ) , λ i ≥ 0 ,

where Vol n stands for the n -dimensional volume and its argument is the Minkowski sum of the scaled convex bodies K i . One can show that f is a homogeneous polynomial of degree n , therefore it can be written as

f ( λ 1 , … , λ r ) = ∑ j 1 , … , j n = 1 r V ( K j 1 , … , K j n ) λ j 1 ⋯ λ j n ,

where the functions V are symmetric. Then V ( K 1 , … , K n ) is called the mixed volume of K 1 , … , K n .

Equivalently,

V ( K 1 , … , K n ) = 1 n ! ∂ n ∂ λ 1 ⋯ ∂ λ n | λ 1 = ⋯ = λ n = + 0 V o l n ( λ 1 K 1 + ⋯ + λ n K n ) .

Properties

The mixed volume is uniquely determined by the following three properties:

1. V ( K , … , K ) = Vol n ( K ) ;
2. V is symmetric in its arguments;
3. V is multilinear: V ( λ K + λ ′ K ′ , K 2 , … , K n ) = λ V ( K , K 2 , … , K n ) + λ ′ V ( K ′ , K 2 , … , K n ) for λ , λ ′ ≥ 0 .

The mixed volume is non-negative and monotonically increasing in each variable: V ( K 1 , K 2 , … , K n ) ≤ V ( K 1 ′ , K 2 , … , K n ) for K 1 ⊆ K 1 ′ .

The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality, are special cases of the Alexandrov–Fenchel inequality.

Quermassintegrals

Let K ⊂ R n be a convex body and let B = B n ⊂ R n be the Euclidean ball of unit radius. The mixed volume

W j ( K ) = V ( K , K , … , K ⏞ n − j  times , B , B , … , B ⏞ j  times )

is called the j-th quermassintegral of K .

The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):

V o l n ( K + t B ) = ∑ j = 0 n ( n j ) W j ( K ) t j .

Intrinsic volumes

The j-th intrinsic volume of K is a different normalization of the quermassintegral, defined by

V j ( K ) = ( n j ) W n − j ( K ) κ n − j , or in other words V o l n ( K + t B ) = ∑ j = 0 n V j ( K ) V o l n − j ( t B n − j ) .

where κ n − j = Vol n − j ( B n − j ) is the volume of the ( n − j ) -dimensional unit ball.

Hadwiger's characterization theorem

Hadwiger's theorem asserts that every valuation on convex bodies in R n that is continuous and invariant under rigid motions of R n is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).