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.
be convex bodies in
-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies
is a homogeneous polynomial of degree
, so can be written as where the functions
For a particular index function
is called the mixed volume of
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): The j-th intrinsic volume of
is a different normalization of the quermassintegral, defined by where
{\displaystyle \kappa _{n-j}={\text{Vol}}_{n-j}(B_{n-j})}
-dimensional unit ball.
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).
(2001) [1994], "Mixed-volume theory", Encyclopedia of Mathematics, EMS Press