Mixed volume

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 the -dimensional space. This number depends on the size and shape of the bodies and on their relative orientation to each other.

Definition[edit]

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. Then is called the mixed volume of .

Equivalently,

Properties[edit]

  • 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[edit]

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[edit]

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[edit]

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[edit]

  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[edit]

Burago, Yu.D. (2001) [1994], "Mixed volume theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4