In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an n-tuple of convex bodies in the n-dimensional space. This number depends on the size of the bodies and their relative positions.
Let K1, K2, ..., Kr be convex bodies in Rn, and consider the function
where Voln stands for the n-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies Ki. One can show that f is a homogeneous polynomial of degree n, therefore it can be written as
where the functions V are symmetric. Then V(T1, ..., Tn) is called the mixed volume of T1, T2, ..., Tn.
- The mixed volume is uniquely determined by the following three properties:
- V(T, ...., T) = Voln(T);
- V is symmetric in its arguments;
- V is multilinear: V(a T + b S, T2, ..., Tn) =a V(T, T2, ..., Tn) + b V(S, T2, ..., Tn) for a,b ≥ 0.
- The mixed volume is non-negative, and increasing in each variable.
- 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.
Let K ⊂ Rn be a convex body, and let B = Bn ⊂ Rn be the Euclidean ball of unit radius. The mixed volume
is called the j-th quermassintegral of K.
The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):
The j-th intrinsic volume of K is a different normalization of the quermassintegral, defined by
- or in other words
where κn−j=Voln−j(Bn−j) is the volume of the (n − j)-dimensional unit ball.
Hadwiger's characterization theorem
Hadwiger's theorem asserts that every valuation on convex bodies in Rn that is continuous and invariant under rigid motions of Rn is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).
- Burago, Yu.D. (2001), "Mixed volume theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
- Klain, D.A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.