Позитивті сызықтық комбинацияларға жабық математикалық жиын

Mathematical set closed under positive linear combinations

In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every A cone need not be convex, or even look like a cone in Euclidean space. When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a positive scalar. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. In this article, only the case of scalars in an ordered field is considered.

A subset C of a vector space V over an ordered field F is a cone (or sometimes called a linear cone) if for each x in C and positive scalar α in F, the product αx is in C. Note that some authors define cone with the scalar α ranging over all non negative scalars (rather than all positive scalars, which does not include 0). A cone C is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.
A cone C is convex if and only if C + C ⊆ C.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C. Some authors use a definition that ensures the origin belongs to C. Because of the scaling parameters α and β, cones are infinite in extent and not bounded. If C is a convex cone, then for any positive scalar α and any x in C the vector It follows that a convex cone C is a special case of a linear cone. It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C is a convex cone if and only if 1=αC = C and 1=C + C = C, for any positive scalar α.

For a vector space V, the empty set, the space V, and any linear subspace of V are convex cones. The conical hull of a finite or infinite set of vectors in is a convex cone. The tangent cones of a convex set are convex cones. The set is a cone but not a convex cone. The norm cone is a convex cone. The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear maps. In particular, if C is a convex cone, so is its opposite and is the largest linear subspace contained in C.
The set of positive semidefinite matrices. The set of nonnegative continuous functions is a convex cone.

An affine convex cone is the set resulting from applying an affine transformation to a convex cone. 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 + C is not a linear cone. However, it is still called an affine convex cone.

A (linear) hyperplane is a set in the form where f is a linear functional on the vector space V. A closed half space is a set in the form or and likewise an open half space uses strict inequality. Half spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex cone C that is not the whole space V must be contained in some closed half space H of V; this is a special case of Farkas' lemma.

A type of cone of particular interest to pure mathematicians is the partially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming.". This object arises when we study cones in together with the lattice A cone is called rational (here we assume "pointed", as defined above) whenever its generators all have integer coordinates, i. e., if is a rational cone, then .

If C is a non empty convex cone in X, then the linear span of C is equal to C C and the largest vector subspace of X contained in C is equal to C ∩ (−C).

A pointed and salient convex cone C induces a partial ordering "≥" on V, defined so that if and only if (If the cone is flat, the same definition gives merely a preorder.) Sums and positive scalar multiples of valid inequalities with respect to this order remain valid inequalities. A vector space with such an order is called an ordered vector space. Examples include the product order on real valued vectors, and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in semidefinite programming.