Mixed volume


In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an of convex bodies in 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

  1. ;
  2. is symmetric in its arguments;
  3. is multilinear: for.
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.