Mixed volume

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https://en.wikipedia.org/wiki/Mixed_volume

In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an -tuple of convex bodies in -dimensional space. This number depends on the size and shape of the bodies and on 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 , therefore it 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:
  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

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

  1. ^ McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi: 10.1007/bf01299276. MR  1089383.
  2. ^ 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) [1994], "Mixed volume theory", Encyclopedia of Mathematics, EMS Press