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. 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).

• The mixed volume is non-negative.

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

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