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

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

$V(K,\dots ,K)={\text{Vol}}_{n}(K)$ ;
$V$ 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 {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 $K$ .^[1]

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})=\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

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).^[2]

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 $\mathrm {Vol} (\pi _{L}(A))$ is equal to $\mathrm {Vol} _{i}(A)$ , 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 ${\begin{aligned}&V_{j}\left(B^{n}\right)={\frac {\kappa _{n}}{\kappa _{n-j}}}{\binom {n}{j}},\quad j=0,\ldots ,n.\\&\kappa _{m}=\operatorname {Vol} _{m}\left(B^{m}\right)={\frac {\pi ^{m/2}}{\Gamma \left({\frac {m}{2}}+1\right)}}\end{aligned}}$ Given an n-dimensional convex body ${\textstyle K}$ , the ${\textstyle j}$ -th intrinsic volume of ${\textstyle K}$ satisfies the Cauchy-Kubota formula^[3] $V_{j}(K):={\frac {\kappa _{n}}{\kappa _{j}\kappa _{n-j}}}{\binom {n}{j}}\int _{\mathrm {G} (n,j)}V_{j}\left(\operatorname {proj} _{E}K\right)\mathrm {d} E$ Here, ${\textstyle \kappa _{j}}$ denotes the ${\textstyle j}$ -dimensional volume of the ${\textstyle j}$ -dimensional unit ball, integration is with respect to the Haar probability measure on ${\textstyle \mathrm {G} (n,j)}$ , the Grassmannian of ${\textstyle j}$ -dimensional subspaces in ${\textstyle \mathbb {R} ^{n}}$ , and ${\textstyle \operatorname {proj} _{E}:\mathbb {R} ^{n}\rightarrow E}$ denotes the orthogonal projection onto ${\textstyle E\in \mathrm {G} (n,j)}$ .