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 ℝn. This number depends on the size and shape of the bodies, and their relative orientation to each other.

## Contents

* 1 Definition
* 2 Properties
* 3 Quermassintegrals
* 3.1 Intrinsic volumes
* 3.2 Hadwiger's characterization theorem
* 3.3 Interpretation
* 4 Examples
* 5 Notes
* 6 External links

## Definition

Let K1,K2,…,Kr be convex bodies in ℝn and consider the function

f(λ1,…,λr)=Voln(λ1K1+⋯+λrKr),λi≥0,

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, so can be written as

f(λ1,…,λr)=∑j1,…,jn=1rV(Kj1,…,Kjn)λj1⋯λjn,

where the functions V are symmetric. For a particular index function j∈{1,…,r}n, the coefficient V(Kj1,…,Kjn) is called the mixed volume of Kj1,…,Kjn.

## Properties

* The mixed volume is uniquely determined by the following three properties:

1. V(K,…,K)=n!Voln(K);
2. V is symmetric in its arguments;
3. V is multilinear: V(λK+λ′K′,K2,…,Kn)=λV(K,K2,…,Kn)+λ′V(K′,K2,…,Kn) for λ,λ′≥0.

* The mixed volume is non-negative and monotonically increasing in each variable: V(K1,K2,…,Kn)≤V(K1′,K2,…,Kn) for K1⊆K1′.
* The Alexandrov-Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

V(K1,K2,K3,…,Kn)≥V(K1,K1,K3,…,Kn)V(K2,K2,K3,…,Kn).

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 K⊂ℝn be a convex body and let B=Bn⊂ℝn be the Euclidean ball of unit radius. The mixed volume

Wj(K)=V(K,K,…,K⏞n−j times,B,B,…,B⏞j times)

is called the j-th quermassintegral of K.[1]

The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):

Voln(K+tB)=∑j=0n(nj)Wj(K)tj.

### Intrinsic volumes

The j-th intrinsic volume of K is a different normalization of the quermassintegral, defined by

Vj(K)=(nj)Wn−j(K)κn−j, or in other words Voln(K+tB)=∑j=0nVj(K)Voln−j(tBn−j)=∑j=0nVj(K)κn−jtn−j.

where κn−j=Voln−j(Bn−j) is the volume of the (n−j)-dimensional unit ball.

### Hadwiger's characterization theorem

Main article: Hadwiger's 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).[2]

### Interpretation

The ith intrinsic volume of a compact convex set A⊆Rn can also be defined in a more geometric way:

If one chooses at random an i-dimensional linear subspace L of Rn and orthogonally projects A onto this subspace L to get πL(A), the expected value of the (Euclidean) i-dimensional volume Vol(πL(A)) is equal to Voli(A), up to a constant factor.

In the case of the two-volume of a three-dimensional convex set, it is a theorem of Cauchy that the expected projection to a random plane is proportional to the surface area.

## Examples

The intrinsic volumes of Bn, the unit ball in ℝn, satisfyVj(Bn)=κnκn−j(nj),j=0,…,n.κm=Volm(Bm)=πm/2Γ(m2+1)Given an n-dimensional convex body K, the j-th intrinsic volume of K satisfies the Cauchy-Kubota formula[3]Vj(K):=κnκjκn−j(nj)∫G(n,j)Vj(projEK)dEHere, κj denotes the j-dimensional volume of the j-dimensional unit ball, integration is with respect to the Haar probability measure on G(n,j), the Grassmannian of j-dimensional subspaces in ℝn, and projE:ℝn->E denotes the orthogonal projection onto E∈G(n,j).

## Notes

1. ↑ McMullen, Peter (1991). "Inequalities between intrinsic volumes". Monatshefte für Mathematik 111 (1): 47-53. doi:10.1007/bf01299276.
2. ↑ Klain, Daniel A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika 42 (2): 329-339. doi:10.1112/s0025579300014625.
3. ↑ Colesanti, Andrea; Ludwig, Monika; Mussnig, Fabian (2025). "The Hadwiger theorem on convex functions, II: Cauchy–Kubota formulas". American Journal of Mathematics 147 (4): 927–955. ISSN 1080-6377. https://muse.jhu.edu/pub/1/article/966289.

## External links

Hazewinkel, Michiel, ed. (2001), "Mixed-volume theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Mixed-volume_theory

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