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Mixed volume

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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

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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

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  • The mixed volume is uniquely determined by the following three properties:
  1. ;
  2. is symmetric in its arguments;
  3. 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

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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

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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

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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

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  1. ^ McMullen, Peter (1991). "Inequalities between intrinsic volumes". Monatshefte für Mathematik. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
  2. ^ Klain, Daniel A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika. 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.
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Burago, Yu.D. (2001) [1994], "Mixed-volume theory", Encyclopedia of Mathematics, EMS Press