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.
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Transcription
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