Mixed volume

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 as well as their relative position.

Definition

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

$f(\lambda _{1},\cdots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r})$

of non-negative λ-s, where Vol_n stands for the n-dimensional volume. One can show that f is a homogeneous polynomial of degree n, therefore it can be written as

$f(\lambda _{1},\cdots ,\lambda _{n})=\sum _{j_{1},\cdots ,j_{n}=1}^{r}V(K_{j_{1}},\cdots ,K_{j_{n}})\lambda _{j_{1}}\cdots \lambda _{j_{n}}~,$

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

Equivalently,

$V(T_{1},\cdots ,T_{n})=\left.{\frac {\partial ^{n}}{\partial t_{1}\cdots \partial t_{n}}}\right|_{t_{1}=\cdots =t_{n}=+0}\mathrm {Vol} _{n}(\lambda _{1}T_{1}+\cdots +\lambda _{n}T_{n})~.$

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:

$V(T_{1},T_{2},T_{3},\cdots ,T_{n})\geq {\sqrt {V(T_{1},T_{1},T_{3},\cdots ,T_{n})V(T_{2},T_{2},T_{3},\cdots ,T_{n})}}~.$

Quermassintegrals

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

$W_{j}(K)=V({\overset {n-j{\text{ times}}}{\overbrace {K,K,\cdots ,K} }},{\overset {j{\text{ times}}}{\overbrace {B,B,\cdots ,B} }})$

is called the j-th quermassintegral of K.

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

$\mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}~.$

Intrinsic volumes

The j-th intrinsic volume of K is defined by

$V_{j}(K)={\binom {n}{j}}{\frac {W_{n-j}(K)}{\kappa _{n-j}}}~,$

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

Hadwiger's characterization theorem

Main article: Hadwiger's theorem

Hadwiger's theorem asserts that every valuation (measure theory) 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).