Mixed volume
None
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
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
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 :
- 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
is called the
j
-th
quermassintegral
of
.
[1]
The definition of mixed volume yields the Steiner formula (named after Jakob Steiner ):
Intrinsic volumes
The
j
-th
intrinsic volume
of
is a different normalization of the quermassintegral, defined by
-
or in other words
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