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An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [8] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.
The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a , the vector ax is in C . [ 2 ]
It uses the concept of a recession cone of a non-empty convex subset S, defined as: = {: +}, where this set is a convex cone containing and satisfying + =. Note that if S is closed and convex then rec S {\displaystyle \operatorname {rec} S} is closed and for all s 0 ∈ S {\displaystyle s_{0}\in S} , rec S = ⋂ t > 0 t ( S − ...
The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign). The dual cone with respect to the Killing form is the maximal invariant convex cone.
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.
In mathematics, especially convex analysis, the recession cone of a set is a cone containing all vectors such that recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.
A convex cone is said to be proper if its closure, also a cone, contains no subspaces. Let C be an open convex cone. Its dual is defined as = {: (,) > ¯}. It is also an open convex cone and C** = C. [2] An open convex cone C is said to be self-dual if C* = C. It is necessarily proper, since it does not contain 0, so cannot contain both X and ...
In linear algebra, a power cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. [1] [2] It is a generalization of the quadratic cone: the quadratic cone is defined using a quadratic equation (with the power 2), whereas a power cone can be defined using any power, not necessarily 2.