Mixed volume

In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an $n$-tuple of convex bodies in the $n$-dimensional space. This number depends on the size of the bodies and on their relative orientation to each other.

Definition

Let ${\displaystyle K_{1},K_{2},\dots ,K_{r}}$ 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 ${\displaystyle {\text{Vol}}_{n}}$ 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 $n$, therefore it 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. Then ${\displaystyle V(K_{1},\dots ,K_{n})}$ is called the mixed volume of ${\displaystyle K_{1},\dots ,K_{n}}$.

Equivalently,

${\displaystyle V(K_{1},\ldots ,K_{n})={\frac {1}{n!}}\left.{\frac {\partial ^{n}}{\partial \lambda _{1}\cdots \partial \lambda _{n}}}\right|_{\lambda _{1}=\cdots =\lambda _{n}=+0}\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{n}K_{n}).}$

Properties

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

1. ${\displaystyle V(K,\dots ,K)={\text{Vol}}_{n}(K)}$;
2. $V$ is symmetric in its arguments;
3. $V$ 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

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

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 ${\mathrm {Vol}}_{n}(K+tB)=\sum _{{j=0}}^{n}V_{j}(K)\,{\mathrm {Vol}}_{{n-j}}(tB_{{n-j}}).$

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

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]

Notes

1. McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
2. Klain, D.A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika. 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.

External links

Burago, Yu.D. (2001), "Mixed volume theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4