Carl Severin Wigert

In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an n-tuple of convex bodies in the n-dimensional space. This number depends on the size of the bodies and their relative positions.

Definition

Let K₁, K₂, ..., K_r be convex bodies in Rⁿ, and consider the function

$f({\lambda }_1,\dots ,{\lambda }_r)={\mathrm{V}\mathrm{o}\mathrm{l}}_n({\lambda }_1{K}_1+\cdots +{\lambda }_r{K}_r),{\lambda }_i\ge 0,$

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

$f({\lambda }_1,\dots ,{\lambda }_r)=\sum _{j_1,\dots ,j_n=1}^{r}V(K_{j_1},\dots ,K_{j_n}){\lambda }_{j_1}\cdots {\lambda }_{j_n},$

where the functions V are symmetric. Then V(T₁, ..., T_n) is called the mixed volume of T₁, T₂, ..., T_n.

Equivalently,

$V(T_1,\dots ,T_n)={\frac{{\partial }^n}{\partial {\lambda }_1\cdots \partial {\lambda }_n}|}_{{\lambda }_1=\cdots ={\lambda }_n=+0}{\mathrm{V}\mathrm{o}\mathrm{l}}_n({\lambda }_1{T}_1+\cdots +{\lambda }_n{T}_n).$

Properties

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

1. V(T, ...., T) = Vol_n(T);
2. V is symmetric in its arguments;
3. V is multilinear: V(a T + b S, T₂, ..., T_n) =a V(T, T₂, ..., T_n) + b V(S, T₂, ..., T_n) for a,b ≥ 0.

• The mixed volume is non-negative, and increasing in each variable.

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

$V(T_1,T_2,T_3,\dots ,T_n)\ge \sqrt{V(T_1,T_1,T_3,\dots ,T_n)V(T_2,T_2,T_3,\dots ,T_n)}.$

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 ⊂ Rⁿ be a convex body, and let B ⊂ Rⁿ be the Euclidean ball. The mixed volume

$W_j(K)=V(\stackrel{n-j\text{ times}}{\overbrace{K,K,\dots ,K}},\stackrel{j\text{ times}}{\overbrace{B,B,\dots ,B}})$

is called the j-th quermassintegral of K.

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

${\mathrm{V}\mathrm{o}\mathrm{l}}_n(K+tB)=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j})W_j(K)t^j.$

Intrinsic volumes

The j-th intrinsic volume of K is defined by

$V_j(K)=(\genfrac{}{}{0ex}{}{n}{j})\frac{W_{n-j}(K)}{{\kappa }_{n-j}},$

where κ_n−j is the volume of the (n − j)-dimensional ball.

Hadwiger's characterization theorem

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

Notes

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