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

Definition
Let $$K_1, K_2, \dots, K_r$$ be convex bodies in $$\mathbb{R}^n$$ 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 $$\text{Vol}_n$$ 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 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. For a particular index function $$ j \in \{1,\ldots,r\}^n $$, the coefficient $$V(K_{j_1}, \dots, K_{j_n})$$ is called the mixed volume of $$K_{j_1}, \dots, K_{j_n}$$.

Properties
V(K, \dots, K) =\text{Vol}_n (K)$$; V(\lambda K + \lambda' K', K_2, \dots, K_n) = \lambda V(K, K_2, \dots, K_n) + \lambda' V(K', K_2, \dots, K_n)$$ for $$ \lambda,\lambda' \geq 0$$.
 * The mixed volume is uniquely determined by the following three properties:
 * 1) $$V$$ is symmetric in its arguments;
 * 2) $$V$$ is multilinear: $$

V(K_1, K_2, \ldots, K_n) \leq V(K_1', K_2, \ldots, K_n)$$ for $$ K_1 \subseteq K_1'$$.
 * The mixed volume is non-negative and monotonically increasing in each variable: $$
 * 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 $$K \subset \mathbb{R}^n$$ be a convex body and let $$B = B_n \subset \mathbb{R}^n$$ 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 $$K$$.

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 $$\kappa_{n-j} = \text{Vol}_{n-j} (B_{n-j})$$ 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 $$\mathbb{R}^n$$ that is continuous and invariant under rigid motions of $$\mathbb{R}^n$$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).