In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, more specifically, in convex geometry

Convex geometry

Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...

, the mixed volume is a way to associate a non-negative number to an n-tuple of convex bodies

Convex body

In mathematics, a convex body in n-dimensional Euclidean space Rn is a compact convex set with non-empty interior.A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point x lies in K if and only if its antipode, −x, also lies in K...

 in the n-dimensional space. This number depends on the size of the bodies and their relative positions.

Definition

Let K₁, K₂, ..., K_r be convex bodies in Rⁿ, and consider the function


of non-negative λ-s, where Vol_n stands for the n-dimensional volume. One can show that f is a homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

 of degree n, therefore it can be written as


where the functions V are symmetric. Then V(T₁, ..., T_n) is called the mixed volume of T₁, T₂, ..., T_n.

Equivalently,

Properties

• The mixed volume is uniquely determined by the following three properties:

1. V(T, ...., T) = Vol_n(T);
2. V is symmetric in its arguments;
3. V is multilinear: V(a T + b S, T₂, ..., T_n) =a V(T, T₂, ..., T_n) + b V(S, T₂, ..., T_n) for a,b ≥ 0.

• The mixed volume is non-negative, and increasing in each variable.

• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov
  Aleksandr Danilovich Aleksandrov
  Aleksandr Danilovich Aleksandrov , and Alexandrov ) , was a Soviet/Russian mathematician, physicist, philosopher and mountaineer.- Scientific career :...
  
   and Werner Fenchel
  Werner Fenchel
  Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory. Fenchel's monographs and lecture-notes were very influential also...
  
  :

Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality

Minkowski's first inequality for convex bodies

In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality....

, are special cases of the Alexandrov–Fenchel inequality.

Quermassintegrals

Let K ⊂ Rⁿ be a convex body, and let B ⊂ Rⁿ be the Euclidean ball. The mixed volume


is called the j-th quermassintegral of K.

The definition of mixed volume yields the Steiner formula (named after Jakob Steiner

Jakob Steiner

Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...

):

Intrinsic volumes

The j-th intrinsic volume of K is defined by


where κ_n−j is the volume of the (n − j)-dimensional ball.

Hadwiger's characterization theorem

Hadwiger's theorem asserts that every valuation (measure theory)

Valuation (measure theory)

In measure theory or at least in the approach to it through domain theory, a valuation is a map from the class of open sets of a topological space to the set positive real numbers including infinity...

on convex bodies in Rⁿ that is continuous and invariant under rigid motions of Rⁿ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).