Mixed volume

In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in. This number depends on the size and shape of the bodies, and 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, so 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:

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.
The definition of mixed volume yields the Steiner formula :

Intrinsic volumes

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

Interpretation

The th intrinsic volume of a compact convex set can also be defined in a more geometric way:
If one chooses at random an -dimensional linear subspace of and orthogonally projects
onto this subspace to get, the expected value of the -dimensional volume is equal to, up to a constant factor.
In the case of the two-volume of a three-dimensional convex set, it is a theorem of Cauchy that the expected projection to a random plane is proportional to the surface area.

Examples

The intrinsic volumes of, the unit ball in, satisfyGiven an n-dimensional convex body, the -th intrinsic volume of satisfies the Cauchy-Kubota formulaHere, denotes the -dimensional volume of the -dimensional unit ball, integration is with respect to the Haar probability measure on, the Grassmannian of -dimensional subspaces in, and denotes the orthogonal projection onto.