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 $\mathbb^n$. 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 $\mathbb^n$ and consider the function

:$f(\lambda_1, \ldots, \lambda_r) = \mathrm_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0,$

where $\text_n$ 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

:$f(\lambda_1, \ldots, \lambda_r) = \sum_^r V(K_, \ldots, K_) \lambda_ \cdots \lambda_,$

where the functions $V$ are symmetric. For a particular index function $j \in \^n$, the coefficient $V(K_, \dots, K_)$ is called the mixed volume of $K_, \dots, K_$.

Properties

* The mixed volume is uniquely determined by the following three properties:
# $V(K, \dots, K) =\text_n (K)$;
# $V$ is symmetric in its arguments;
# $V$ is multilinear: $V(\lambda K + \lambda' K', K_2, \dots, K_n) = \lambda V(K, K_2, \dots, K_n) + \lambda' V(K', K_2, \dots, K_n)$ for $\lambda,\lambda' \geq 0$.

* The mixed volume is 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:

::$V(K_1, K_2, K_3, \ldots, K_n) \geq \sqrt.$

: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 $K \subset \mathbb^n$ be a convex body and let $B = B_n \subset \mathbb^n$ be the Euclidean ball of unit radius. The mixed volume

:$W_j(K) = V(\overset, \overset)$

is called the ''j''-th quermassintegral of $K$.

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

:$\mathrm_n(K + tB) = \sum_^n \binom W_j(K) t^j.$

Intrinsic volumes

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

:$V_j(K) = \binom \frac,$ or in other words $\mathrm_n(K + tB) = \sum_^n V_j(K)\, \mathrm_(tB_) = \sum_^n V_j(K)\,\kappa_t^.$

where $\kappa_ = \text_ (B_)$ 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 $\mathbb^n$ that is continuous and invariant under rigid motions of $\mathbb^n$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).

Notes

External links

{{eom, id=Mixed-volume_theory, title=Mixed-volume theory, first=Yu.D., last=Burago

Convex geometry
Integral geometry