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
. This number depends on the size and shape of the bodies, and their relative orientation to each other.
Definition
Let
be convex bodies in
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de52e17162a91b5e618d5a91301bda714657dbed)
where
stands for the
-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies
. One can show that
is a homogeneous polynomial of degree
, 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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f0f0ebee7a56fb22f95867257168e8331929065)
where the functions
are symmetric. For a particular index function
, the coefficient
is called the mixed volume of
.
Properties
- The mixed volume is uniquely determined by the following three properties:
;
is symmetric in its arguments;
is multilinear:
for
.
- The mixed volume is non-negative and monotonically increasing in each variable:
for
. - 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})}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf89dee63c4414c546760f8d3e0a424f5dae142)
- 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
be a convex body and let
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} }})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4aeaeb5d651c5571fc20ded2c5e2ae858a01e8)
is called the j-th quermassintegral of
.[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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eed268a9c70f484b76aca321f397db60b7e9b731)
Intrinsic volumes
The j-th intrinsic volume of
is a different normalization of the quermassintegral, defined by
or in other words ![{\displaystyle \mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}V_{j}(K)\,\mathrm {Vol} _{n-j}(tB_{n-j}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec957640e4f5681c8e50a7ace749c3aa88ee36a7)
where
is the volume of the
-dimensional unit ball.
Hadwiger's characterization theorem
Hadwiger's theorem asserts that every valuation on convex bodies in
that is continuous and invariant under rigid motions of
is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).[2]
Notes
- ^ McMullen, Peter (1991). "Inequalities between intrinsic volumes". Monatshefte für Mathematik. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
- ^ Klain, Daniel 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) [1994], "Mixed-volume theory", Encyclopedia of Mathematics, EMS Press