In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''C'' of a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is said to be absolutely convex or disked if it is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and
balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull or the absolute convex hull of a set is the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of all disks containing that set.
Definition
A subset
of a real or complex vector space
is called a ' and is said to be ', ', and ' if any of the following equivalent conditions is satisfied:
- is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and balanced set
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \ ...
.
- for any scalar and if then
- for all scalars and if then
- for any scalars and if then
- for any scalars if then
The smallest
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
(respectively,
balanced) subset of
containing a given set is called the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
(respectively, the balanced hull) of that set and is denoted by
(respectively,
).
Similarly, the ', the ', and the ' of a set
is defined to be the smallest disk (with respect to subset
inclusion) containing
The disked hull of
will be denoted by
or
and it is equal to each of the following sets:
- which is the convex hull of the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, \l ...
of ; thus,
* In general, is possible, even in finite dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, ยง2.36 It is sometimes called Hamel dimension (after ...
vector spaces.
- the intersection of all disks containing
Sufficient conditions
The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however,
unions of absolutely convex sets need not be absolutely convex anymore.
If
is a disk in
then
is absorbing in
if and only if
Properties
If
is an
absorbing disk in a vector space
then there exists an absorbing disk
in
such that
If
is a disk and
and
are scalars then
and
The absolutely convex hull of a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of m ...
in a locally convex topological vector space is again bounded.
If
is a bounded disk in a TVS
and if
is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
in
then the partial sums
are
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, where for all
In particular, if in addition
is a
sequentially complete
In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in .
is called sequentially complete if i ...
subset of
then this series
converges in
to some point of
The convex balanced hull of
contains both the convex hull of
and the balanced hull of
Furthermore, it contains the balanced hull of the convex hull of
thus
where the example below shows that this inclusion might be strict.
However, for any subsets
if
then
which implies
Examples
Although
the convex balanced hull of
is necessarily equal to the balanced hull of the convex hull of
For an example where
let
be the real vector space
and let
Then
is a strict subset of
that is not even convex;
in particular, this example also shows that the balanced hull of a convex set is necessarily convex.
The set
is equal to the closed and filled square in
with vertices
and
(this is because the balanced set
must contain both
and
where since
is also convex, it must consequently contain the solid square
which for this particular example happens to also be balanced so that
). However,
is equal to the horizontal closed line segment between the two points in
so that
is instead a closed "
hour glass
An hourglass (or sandglass, sand timer, sand clock or egg timer) is a device used to measure the passage of time. It comprises two glass bulbs connected vertically by a narrow neck that allows a regulated flow of a substance (historically san ...
shaped" subset that intersects the
-axis at exactly the origin and is the union of two closed and filled
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s: one whose vertices are the origin together with
and the other triangle whose vertices are the origin together with
This non-convex filled "hour-glass"
is a proper subset of the filled square
Generalizations
Given a fixed real number
a is any subset
of a vector space
with the property that
whenever
and
are non-negative scalars satisfying
It is called an or a if
whenever
and
are scalars satisfying
A is any non-negative function
that satisfies the following conditions:
#
Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
/
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
:
for all
#
Absolute homogeneity of degree :
for all
and all scalars
This generalizes the definition of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s since a map is a seminorm if and only if it is a
-seminorm (using
).
There exist
-seminorms that are not
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
s. For example, whenever
then the map
used to define the
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
is a
-seminorm but not a seminorm.
Given
a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is (meaning that its topology is induced by some
-seminorm) if and only if it has a
bounded -convex neighborhood of the origin.
See also
*
*
*
*
*
*
*
* , for vectors in physics
*
References
Bibliography
*
*
*
*
{{Convex analysis and variational analysis
Abstract algebra
Convex analysis
Convex geometry
Group theory
Linear algebra