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

Definition

Let ${\displaystyle K_{1},K_{2},\dots ,K_{r}}$ be convex bodies in ${\displaystyle \mathbb {R} ^{n}}$ and consider the function

${\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r}),\qquad \lambda _{i}\geq 0,}$

where ${\displaystyle {\text{Vol}}_{n}}$ stands for the ${\displaystyle n}$-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies ${\displaystyle K_{i}}$. One can show that ${\displaystyle f}$ is a homogeneous polynomial of degree ${\displaystyle n}$, so can be written as

${\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\sum _{j_{1},\ldots ,j_{n}=1}^{r}V(K_{j_{1}},\ldots ,K_{j_{n}})\lambda _{j_{1}}\cdots \lambda _{j_{n}},}$

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

Properties

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

1. ${\displaystyle V(K,\dots ,K)=n!{\text{Vol}}_{n}(K)}$;
2. ${\displaystyle V}$ is symmetric in its arguments;
3. ${\displaystyle V}$ is multilinear: ${\displaystyle V(\lambda K+\lambda 'K',K_{2},\dots ,K_{n})=\lambda V(K,K_{2},\dots ,K_{n})+\lambda 'V(K',K_{2},\dots ,K_{n})}$ for ${\displaystyle \lambda ,\lambda '\geq 0}$.

• The mixed volume is non-negative and monotonically increasing in each variable: ${\displaystyle V(K_{1},K_{2},\ldots ,K_{n})\leq V(K_{1}',K_{2},\ldots ,K_{n})}$ for ${\displaystyle K_{1}\subseteq K_{1}'}$.
• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

${\displaystyle V(K_{1},K_{2},K_{3},\ldots ,K_{n})\geq {\sqrt {V(K_{1},K_{1},K_{3},\ldots ,K_{n})V(K_{2},K_{2},K_{3},\ldots ,K_{n})}}.}$

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

${\displaystyle W_{j}(K)=V({\overset {n-j{\text{ times}}}{\overbrace {K,K,\ldots ,K} }},{\overset {j{\text{ times}}}{\overbrace {B,B,\ldots ,B} }})}$

is called the j-th quermassintegral of ${\displaystyle K}$.^[1]

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

${\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}.}$

Intrinsic volumes

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

${\displaystyle V_{j}(K)={\binom {n}{j}}{\frac {W_{n-j}(K)}{\kappa _{n-j}}},}$ or in other words ${\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}V_{j}(K)\,\mathrm {Vol} _{n-j}(tB_{n-j})=\sum _{j=0}^{n}V_{j}(K)\,\kappa _{n-j}t^{n-j}.}$

where ${\displaystyle \kappa _{n-j}={\text{Vol}}_{n-j}(B_{n-j})}$ is the volume of the ${\displaystyle (n-j)}$-dimensional unit ball.

Hadwiger's characterization theorem

Main article: Hadwiger's theorem

Hadwiger's theorem asserts that every valuation on convex bodies in ${\displaystyle \mathbb {R} ^{n}}$ that is continuous and invariant under rigid motions of ${\displaystyle \mathbb {R} ^{n}}$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).^[2]

Interpretation

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

If one chooses at random an ${\displaystyle i}$-dimensional linear subspace ${\displaystyle L}$ of ${\displaystyle R^{n}}$ and orthogonally projects ${\displaystyle A}$ onto this subspace ${\displaystyle L}$ to get ${\displaystyle \pi _{L}(A)}$, the expected value of the (Euclidean) ${\displaystyle i}$-dimensional volume ${\displaystyle \mathrm {Vol} (\pi _{L}(A))}$ is equal to ${\displaystyle \mathrm {Vol} _{i}(A)}$, 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 ${\displaystyle B^{n}}$, the unit ball in ${\displaystyle \mathbb {R} ^{n}}$, satisfy$${\displaystyle {\begin{aligned}&V_{j}\left(B^{n}\right)={\frac {\kappa _{n}}{\kappa _{n-j}}}{\binom {n}{j}},\quad j=0,\ldots ,n.\\&\kappa _{m}=\operatorname {Vol} _{m}\left(B^{m}\right)={\frac {\pi ^{m/2}}{\Gamma \left({\frac {m}{2}}+1\right)}}\end{aligned}}}$$Given an n-dimensional convex body ${\textstyle K}$, the ${\textstyle j}$-th intrinsic volume of ${\textstyle K}$ satisfies the Cauchy-Kubota formula^[3]$${\displaystyle V_{j}(K):={\frac {\kappa _{n}}{\kappa _{j}\kappa _{n-j}}}{\binom {n}{j}}\int _{\mathrm {G} (n,j)}V_{j}\left(\operatorname {proj} _{E}K\right)\mathrm {d} E}$$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 ${\textstyle E\in \mathrm {G} (n,j)}$.