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.
Definitionedit
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 .
Propertiesedit
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 volumesedit
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 theoremedit
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]
Notesedit
^McMullen, Peter (1991). "Inequalities between intrinsic volumes". Monatshefte für Mathematik. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
^Klain, Daniel A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika. 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.