Definition
A subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.
The defining condition can be written more succinctly as "αC + βC = C" for any positive scalars α, β.
The 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.
The empty set, the space V, and any linear subspace of V (including the trivial subspace {0}) are convex cones by this definition. Other examples are the set of all positive multiples of an arbitrary vector v of V, or the positive orthant of (the set of all vectors whose coordinates are all positive).
A more general example is the set of all vectors λx such that λ is a positive scalar and x is an element of some convex subset X of V. In particular, if V is a normed vector space, and X is an open (resp. closed) ball of V that does not contain 0, this construction gives an open (resp. closed) convex circular 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 -C; and C(-C) is the largest linear subspace contained in C.
Read more about this topic: Convex Cone
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)