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A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
The polar cone of a convex cone is the set := { : , } This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x ∈ X {\displaystyle x\in X} is the locus { y : y , x = 0 } {\displaystyle \{y~:~\langle y,x\rangle =0\}} ; the dual ...
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel ...
A set C (blue) and its dual cone C * (red).. A duality in geometry is provided by the dual cone construction. Given a set of points in the plane (or more generally points in ), the dual cone is defined as the set consisting of those points (,) satisfying + for all points (,) in , as illustrated in the diagram.
A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface.
The pole is the point, the polar the line. See Pole and polar. By calculation one checks the following properties of the pole-polar relation of the hyperbola: For a point (pole) on the hyperbola the polar is the tangent at this point (see diagram: , ).
The set of continuous linear functionals ℓ for which σ K (ℓ) ≤ 1 is known as the polar set of K. The set of continuous linear functionals ℓ for which σ K (ℓ) ≤ 0 is known as the (negative) polar cone of K. Clearly, both the polar set and the negative polar cone are subsets of the barrier cone.
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