In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a valuation is a finitely additive function from a collection of subsets of a set
to 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 (just notation, not necessarily th ...
.
For example,
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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
is a valuation on finite unions of
convex bodies of
Other examples of valuations on finite unions of convex bodies of
are
surface area
The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
,
mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
width, and
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's ...
.
In geometry,
continuity (or
smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
) conditions are often 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 ...
s and in particular
Hilbert's third problem
The third of Hilbert's problems, 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 polyhedron, polyhedra of equal volume, is it always possible t ...
, which has grown into a rich theory reliant on tools from abstract algebra.
Definition
Let
be a set, and let
be a collection of 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 of
are
* the
convex bodies in
*
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
convex polytopes
''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, M ...
in
*
convex cones
* smooth compact polyhedra in a smooth manifold
Let
be the set of convex bodies in
Then some valuations on
are
* 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's ...
*
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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
restricted to
*
intrinsic volume (and, more generally,
mixed volume)
* the map
where
is the
support function of
Some other valuations are
* the lattice point enumerator
, where
is a lattice polytope
*
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, on the family of finite sets
Valuations on convex bodies
From here on, let
, let
be the set of convex bodies in
, and let
be a valuation on
.
We say
is ''translation invariant'' if, for all
and
, we have
.
Let
. The
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 set, non-empty compact space, compact subsets o ...
is defined as
where
is the
-neighborhood of
under some Euclidean inner product. 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 ...
.
The space of continuous, translation-invariant valuations from
to
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 (, ) 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 vectors and ...
.
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
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Biography
Although born in Karlsruhe, Ge ...
proved that
(where
) coincides with the
-dimensional space of Lebesgue measures on
A valuation
is ''simple'' if
for all convex bodies with