In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set
with values in an abelian
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
.
For example, the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space
Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
.
In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
s and in particular
Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...
, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra.
Definition
Let
be a set and
be a collection of admissible subsets of
A function
on
with values in an abelian semigroup
is called a valuation if it satisfies
whenever
and
are elements of
If
then one always assumes
Examples
Some common examples for the admissible sets
are non-empty compact convex sets (convex bodies) in
compact convex polytopes in
convex cones, and smooth compact polyhedra in a smooth manifold
Let
be a finite-dimensional vector space over
and let
denote the set of convex bodies in
* The
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
is a valuation on
and it extends as a valuation to the collection of finite unions of convex bodies.
* Any
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
on
restricted to convex bodies, is a valuation on
* Among the valuations derived from volume are the
intrinsic volumes,
where
is a normalizing constant and
is the Euclidean unit ball, and more generally the
mixed volumes (with some entries fixed arbitrarily).
* The lattice point enumerator
where
is the integer lattice, is a valuation on lattice polytopes.
* The map
where
is the
support function
In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n
describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n.
Any n ...
of
is a valuation on
Valuations on convex bodies
A valuation on
is said to be ''translation invariant'' if
for all
and all convex bodies
Let
be two convex bodies in
After a choice of Euclidean inner product, their
Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric ...
is defined by
where
denotes the
-neighborhood of
Equipped with this metric,
is a
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
.
The space of continuous, translation-invariant valuations on
taking values in the complex numbers
is denoted by
The topology on
is the topology of uniform convergence on compact subsets of
Equipped with the norm
where
is a bounded subset with nonempty interior,
is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
Homogeneous valuations
A translation-invariant continuous valuation
is said to be ''
-homogeneous'' if
for all
and
The subset
of
-homogeneous valuations is a vector subspace of
McMullen's decomposition theorem
states that
In particular, the degree of a homogeneous valuation is always an integer between
and
Valuations are not only graded by the degree of homogeneity, but also by the parity with respect to the reflection through the origin, namely
where
with
if and only if
for all convex bodies
The elements of
and
are said to be ''even'' and ''odd'', respectively.
It is a simple fact that
is
-dimensional and spanned by the Euler characteristic
that is, consists of the constant valuations on
In 1957
Hadwiger proved that
(where
) coincides with the
-dimensional space of Lebesgue measures on
A valuation
is ''simple'' if
for all convex bodies with