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 supporting hyperplane of a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

S

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

\mathbb R^n

is a

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

that has both of the following two properties: *

S

is entirely contained in one of the two closed half-spaces bounded by the hyperplane, *

S

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 and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

states that if

S

is a

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

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 ...

X=\mathbb^n,

and

x_0

is a point on the boundary of

S,

then there exists a supporting hyperplane containing

x_0.

x^* \in X^* \backslash \

(

X^*

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 cons ...

X

x^*

is a nonzero linear functional) such that

x^*\left(x_0\right) \geq x^*(x)

for all

x \in S

, then :

H = \

defines a supporting hyperplane. Conversely, if

S

is a

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

with nonempty interior such that every point on the boundary has a supporting hyperplane, then

S

is a convex set, and is the intersection of all its supporting closed half-spaces. The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set

S

is not convex, the statement of the theorem is not true at all points on the boundary of

S,

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. The forward direction can be proved as a special case of the separating hyperplane theorem (see the page for the proof). For the converse direction,

Notes

References & further reading

* * * *{{cite book , last = Soltan , first = V. , title = Support and separation properties of convex sets in finite dimension , publisher = Extracta Math. Vol. 36, no. 2, 241-278 , year = 2021 Convex geometry Functional analysis Duality theories

Supporting hyperplane theorem

See also

Notes

References & further reading