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

$f({\lambda }_1,\dots ,{\lambda }_r)={\mathrm{V}\mathrm{o}\mathrm{l}}_n({\lambda }_1{K}_1+\cdots +{\lambda }_r{K}_r),{\lambda }_i\ge 0,$

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

$f({\lambda }_1,\dots ,{\lambda }_r)=\sum _{j_1,\dots ,j_n=1}^{r}V(K_{j_1},\dots ,K_{j_n}){\lambda }_{j_1}\cdots {\lambda }_{j_n},$

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

Properties

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

1. $V(K,\dots ,K)=n!{\text{Vol}}_n(K)$;
2. $V$ is symmetric in its arguments;
3. $V$ is multilinear: $V(\lambda K+{\lambda }^{\prime }{K}^{\prime },K_2,\dots ,K_n)=\lambda V(K,K_2,\dots ,K_n)+{\lambda }^{\prime }V(K^{\prime },K_2,\dots ,K_n)$ for $\lambda ,{\lambda }^{\prime }\ge 0$.

• The mixed volume is non-negative and monotonically increasing in each variable: $V(K_1,K_2,\dots ,K_n)\le V({K_1}^{\prime },K_2,\dots ,K_n)$ for $K_1\subseteq {K_1}^{\prime }$.
• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

$V(K_1,K_2,K_3,\dots ,K_n)\ge \sqrt{V(K_1,K_1,K_3,\dots ,K_n)V(K_2,K_2,K_3,\dots ,K_n)}.$

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

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

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

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

${\mathrm{V}\mathrm{o}\mathrm{l}}_n(K+tB)=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j})W_j(K)t^j.$

Intrinsic volumes

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

$V_j(K)=(\genfrac{}{}{0ex}{}{n}{j})\frac{W_{n-j}(K)}{{\kappa }_{n-j}},$ or in other words ${\mathrm{V}\mathrm{o}\mathrm{l}}_n(K+tB)=\sum _{j=0}^n{V}_j(K){\mathrm{V}\mathrm{o}\mathrm{l}}_{n-j}(t{B}_{n-j})=\sum _{j=0}^n{V}_j(K){\kappa }_{n-j}{t}^{n-j}.$

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

Main article: Hadwiger's 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).

Interpretation

The $i$th intrinsic volume of a compact convex set $A\subseteq R^n$ can also be defined in a more geometric way:

If one chooses at random an $i$-dimensional linear subspace $L$ of $R^n$ and orthogonally projects $A$ onto this subspace $L$ to get ${\pi }_L(A)$, the expected value of the (Euclidean) $i$-dimensional volume $\mathrm{V}\mathrm{o}\mathrm{l}({\pi }_L(A))$ is equal to ${\mathrm{V}\mathrm{o}\mathrm{l}}_i(A)$, up to a constant factor.

In the case of the two-volume of a three-dimensional convex set, it is a theorem of Cauchy that the expected projection to a random plane is proportional to the surface area.

Examples

The intrinsic volumes of $B^n$, the unit ball in ${\mathbb{ℝ}}^n$, satisfy$$\begin{array}{rl}& V_j\left(B^n\right)=\frac{{\kappa }_n}{{\kappa }_{n-j}}(\genfrac{}{}{0ex}{}{n}{j}),j=0,\dots ,n.\\ & {\kappa }_m={\mathrm{Vol}}_m\left(B^m\right)=\frac{{\pi }^{m/2}}{\mathrm{\Gamma }(\frac{m}{2}+1)}\end{array}$$Given an n-dimensional convex body $K$, the $j$-th intrinsic volume of $K$ satisfies the Cauchy-Kubota formula$$V_j(K):=\frac{{\kappa }_n}{{\kappa }_j{\kappa }_{n-j}}(\genfrac{}{}{0ex}{}{n}{j})\underset{\mathrm{G}(n,j)}{\int }{V}_j\left({\mathrm{proj}}_{E}K\right)\mathrm{d}E$$Here, ${\kappa }_j$ denotes the $j$-dimensional volume of the $j$-dimensional unit ball, integration is with respect to the Haar probability measure on $\mathrm{G}(n,j)$, the Grassmannian of $j$-dimensional subspaces in ${\mathbb{ℝ}}^n$, and ${\mathrm{proj}}_E:{\mathbb{ℝ}}^n\to E$ denotes the orthogonal projection onto $E\in \mathrm{G}(n,j)$.

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