Mixed volume

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.^[1]

Definition

Let K₁, K₂, ..., K_r be convex bodies in Rⁿ, and consider the function

$f(\lambda_1, \ldots, \lambda_r) = \mathrm{Vol}_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0,$

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, therefore it can be written as

$f(\lambda_1, \ldots, \lambda_r) = \sum_{j_1, \ldots, j_n = 1}^r V(K_{j_1}, \ldots, K_{j_n}) \lambda_{j_1} \cdots \lambda_{j_n},$

where the functions V are symmetric. Then V(T₁, ..., T_n) is called the mixed volume of T₁, T₂, ..., T_n.

Equivalently,

$V(T_1, \ldots, T_n) = \left. \frac{\partial^n}{\partial \lambda_1 \cdots \partial \lambda_n}\right|_{\lambda_1 = \cdots = \lambda_n = +0} \mathrm{Vol}_n(\lambda_1 T_1 + \cdots + \lambda_n T_n).$

Properties

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

1. V(T, ...., T) = Vol_n(T);
2. V is symmetric in its arguments;
3. V is multilinear: V(a T + b S, T₂, ..., T_n) =a V(T, T₂, ..., T_n) + b V(S, T₂, ..., T_n) 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:

$V(T_1, T_2, T_3, \ldots, T_n) \geq \sqrt{V(T_1, T_1, T_3, \ldots, T_n) V(T_2,T_2, T_3,\ldots,T_n)}.$

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

$W_j(K) = V(\overset{n-j \text{ times}}{\overbrace{K,K, \ldots,K}}, \overset{j \text{ times}}{\overbrace{B,B,\ldots,B}})$

is called the j-th quermassintegral of K.^[2]

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

$\mathrm{Vol}_n(K + tB) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j.$

Intrinsic volumes

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

$V_j(K) = \binom{n}{j} \frac{W_{n-j}(K)}{\kappa_{n-j}},$ or in other words $\mathrm{Vol}_n(K + tB) = \sum_{j=0}^n V_j(K)\, \mathrm{Vol}_{n-j}(tB_{n-j}).$

where κ_n−j=Vol_n−j(B_n−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).^[3]

Notes

1. Burago, Yu.D. (2001), "Mixed volume theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
2. McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
3. Klain, D.A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.