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 supporting hyperplane of a
set in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
is a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
that has both of the following two properties:
*
is entirely contained in one of the two
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
half-spaces bounded by the hyperplane,
*
has at least one boundary-point on the hyperplane.
Here, a closed half-space is the half-space that includes the points within the hyperplane.
Supporting hyperplane theorem
This
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
states that if
is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
in the
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 ...
and
is a point on the
boundary of
then there exists a supporting hyperplane containing
If
(
is the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of
,
is a nonzero linear functional) such that
for all
, then
:
defines a supporting hyperplane.
Conversely, if
is a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
with nonempty
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
such that every point on the boundary has a supporting hyperplane, then
is a convex set.
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set
is not convex, the statement of the theorem is not true at all points on the boundary of
as illustrated in the third picture on the right.
The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.
[ Cassels, John W. S. (1997), ''An Introduction to the Geometry of Numbers'', Springer Classics in Mathematics (reprint of 1959 and 1971 Springer-Verlag ed.), Springer-Verlag.]
A related result is the
separating hyperplane theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least o ...
, that every two disjoint convex sets can be separated by a hyperplane.
See also
*
Support function
*
Supporting line (supporting hyperplanes in
)
Notes
References & further reading
*
*
*{{cite book
, last = Goh
, first = C. J.
, author2=Yang, X.Q.
, title = Duality in optimization and variational inequalities
, publisher = London; New York: Taylor & Francis
, year = 2002
, isbn = 0-415-27479-6
, page = 13
Convex geometry
Functional analysis
Duality theories