# Mixed volume

*https://en.wikipedia.org/wiki/Mixed_volume*

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

## Definition

Let be convex bodies in and consider the function

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

where the functions are symmetric. For a particular index function , the coefficient is called the mixed volume of .

## Properties

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

- ;
- is symmetric in its arguments;
- is multilinear: for .

- The mixed volume is non-negative and monotonically increasing in each variable: for .
- 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 be a convex body and let be the Euclidean ball of unit radius. The mixed volume

is called the *j*-th **quermassintegral** of .^{
[1]}

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

### Intrinsic volumes

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

- or in other words

where is the volume of the -dimensional unit ball.

### Hadwiger's characterization theorem

Hadwiger's theorem asserts that every
valuation on convex bodies in that is continuous and invariant under rigid motions of is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).^{
[2]}

## Notes

**^**McMullen, P. (1991). "Inequalities between intrinsic volumes".*Monatsh. Math*.**111**(1): 47–53. doi: 10.1007/bf01299276. MR 1089383.**^**Klain, D.A. (1995). "A short proof of Hadwiger's characterization theorem".*Mathematika*.**42**(2): 329–339. doi: 10.1112/s0025579300014625. MR 1376731.

## External links

Burago, Yu.D. (2001) [1994],
"Mixed volume theory", *
Encyclopedia of Mathematics*,
EMS Press