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 and their relative positions.

Definition

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

$f(\lambda _{1},\ldots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r}),\qquad \lambda _{i}\geq 0,$

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

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

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

Equivalently,

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

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:

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

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

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

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 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).