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 [math]\displaystyle{ \mathbb{R}^n }[/math]. This number depends on the size and shape of the bodies, and their relative orientation to each other.

Definition

Let [math]\displaystyle{ K_1, K_2, \dots, K_r }[/math] be convex bodies in [math]\displaystyle{ \mathbb{R}^n }[/math] and consider the function

[math]\displaystyle{ f(\lambda_1, \ldots, \lambda_r) = \mathrm{Vol}_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0, }[/math]

where [math]\displaystyle{ \text{Vol}_n }[/math] stands for the [math]\displaystyle{ n }[/math]-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies [math]\displaystyle{ K_i }[/math]. One can show that [math]\displaystyle{ f }[/math] is a homogeneous polynomial of degree [math]\displaystyle{ n }[/math], so can be written as

[math]\displaystyle{ 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}, }[/math]

where the functions [math]\displaystyle{ V }[/math] are symmetric. For a particular index function [math]\displaystyle{ j \in \{1,\ldots,r\}^n }[/math], the coefficient [math]\displaystyle{ V(K_{j_1}, \dots, K_{j_n}) }[/math] is called the mixed volume of [math]\displaystyle{ K_{j_1}, \dots, K_{j_n} }[/math].

Properties

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

1. [math]\displaystyle{ V(K, \dots, K) =\text{Vol}_n (K) }[/math];
2. [math]\displaystyle{ V }[/math] is symmetric in its arguments;
3. [math]\displaystyle{ V }[/math] is multilinear: [math]\displaystyle{ V(\lambda K + \lambda' K', K_2, \dots, K_n) = \lambda V(K, K_2, \dots, K_n) + \lambda' V(K', K_2, \dots, K_n) }[/math] for [math]\displaystyle{ \lambda,\lambda' \geq 0 }[/math].

• The mixed volume is non-negative and monotonically increasing in each variable: [math]\displaystyle{ V(K_1, K_2, \ldots, K_n) \leq V(K_1', K_2, \ldots, K_n) }[/math] for [math]\displaystyle{ K_1 \subseteq K_1' }[/math].
• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

[math]\displaystyle{ 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)}. }[/math]

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 [math]\displaystyle{ K \subset \mathbb{R}^n }[/math] be a convex body and let [math]\displaystyle{ B = B_n \subset \mathbb{R}^n }[/math] be the Euclidean ball of unit radius. The mixed volume

[math]\displaystyle{ W_j(K) = V(\overset{n-j \text{ times}}{\overbrace{K,K, \ldots,K}}, \overset{j \text{ times}}{\overbrace{B,B,\ldots,B}}) }[/math]

is called the j-th quermassintegral of [math]\displaystyle{ K }[/math].

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

[math]\displaystyle{ \mathrm{Vol}_n(K + tB) = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j. }[/math]

Intrinsic volumes

The j-th intrinsic volume of [math]\displaystyle{ K }[/math] is a different normalization of the quermassintegral, defined by

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

where [math]\displaystyle{ \kappa_{n-j} = \text{Vol}_{n-j} (B_{n-j}) }[/math] is the volume of the [math]\displaystyle{ (n-j) }[/math]-dimensional unit ball.

Hadwiger's characterization theorem

Hadwiger's theorem asserts that every valuation on convex bodies in [math]\displaystyle{ \mathbb{R}^n }[/math] that is continuous and invariant under rigid motions of [math]\displaystyle{ \mathbb{R}^n }[/math] is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).

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