Mixed volume explained

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

R ^{n} . This number depends on the size and shape of the bodies, and their relative orientation to each other.

Definition

Let

K _{1,} K _{2,} ... , K _{r} be convex bodies in R ^{n} and consider the functionf( λ _{1,} \ldots, λ _{r)} = Vol _{n} ( λ _{1} K _{1} + … + λ _{r} K _{r),} λ _{i} \geq 0 ,

where

Vol _{n} stands for the n -dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies K _{i} . One can show that f is a homogeneous polynomial of degree n , so can be written asf( λ _{1,} \ldots, λ _{r)} = \sum ^{r} _{j _{1,} \ldots, j _{n} =1} V(K _{j _{1}} , \ldots, K _{j _{n}} ) λ _{j _{1}} … λ _{j _{n}} ,

where the functions

V are symmetric. For a particular index function j \in \{ 1 ,\ldots,r\} ^{n} , the coefficient V(K _{j _{1}} , ... , K _{j _{n}} ) is called the mixed volume of K _{j _{1}} , ... , K _{j _{n}} .

Properties

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

V(K, ... , K) = Vol _{n} (K) ;V is symmetric in its arguments;V is multilinear: V( λ K + λ ' K', K _{2,} ... , K _{n)} = λ V(K, K _{2,} ... , K _{n) +} λ ' V(K', K _{2,} ... , K _{n)} for λ , λ ' \geq 0 .

• The mixed volume is non-negative and monotonically increasing in each variable:

V(K _{1,} K _{2,} \ldots, K _{n)} \leq V(K _{1',} K _{2,} \ldots, K _{n)} for K _{1} \subseteq K _{1'} .

• The Alexandrov - Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

V(K _{1,} K _{2,} K _{3,} \ldots, K _{n)} \geq \sqrt{V(K _{1,} K _{1,} K _{3,} \ldots, K _{n)} V(K _{2,K} _{2,} K _{3,\ldots,K} _{n)}.}

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 \subset R ^{n} be a convex body and let B = B _{n} \subset R ^{n} be the Euclidean ball of unit radius. The mixed volumeW _{j(K)} = V(\overset{n - j times }, \overset)

is called the j-th quermassintegral of

K .^[1]

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

Vol _{n(K} + tB) = \sum ^{n} _{j =0} \binom{n}{j} W _{j(K)} t ^{j.}

Intrinsic volumes

The j-th intrinsic volume of

K is a different normalization of the quermassintegral, defined byV _{j(K)} = \binom{n}{j} (W _{n-j} (K))/(\kappa _{n-j}) , or in other words Vol _{n(K} + tB) = \sum ^{n} _{j =0} V _{j(K)} Vol _{n-j} (tB _{n-j} ).

where

\kappa _{n-j} = Vol _{n-j} (B _{n-j} ) is the volume of the (n - j) -dimensional unit ball.

Hadwiger's characterization theorem

See main article: Hadwiger's theorem.

Hadwiger's theorem asserts that every valuation on convex bodies in

R ^{n} that is continuous and invariant under rigid motions of R ^{n} is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).^[2]

Notes and References

1. 1089383. McMullen. Peter. Peter McMullen. Inequalities between intrinsic volumes. Monatshefte für Mathematik. 111. 1991. 1. 47 - 53. 10.1007/bf01299276. free.
2. 1376731. Klain. Daniel A.. A short proof of Hadwiger's characterization theorem. Mathematika. 42. 1995. 2. 329 - 339. 10.1112/s0025579300014625.