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 $K_{1},K_{2},\dots ,K_{r}$ 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 $n$ , 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 $V(K_{1},\dots ,K_{n})$ is called the mixed volume of $K_{1},\dots ,K_{n}$ .

Equivalently,

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:

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

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).^[2]

Notes

↑ McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
↑ 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

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