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.

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