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:

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 . 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). Notes

External links

Convex geometry
Integral geometry