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.
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 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]