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Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient (or strictly convex) otherwise.
In fact, it is the intersection of all convex cones containing S plus the origin. [1] If S is a compact set (in particular, when it is a finite non-empty set of points), then the condition "plus the origin" is unnecessary.
That is, cones through which all other cones factor. A cone φ from L to F is a universal cone if for any other cone ψ from N to F there is a unique morphism from ψ to φ. Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in C J), or a terminal object in (Δ ↓ F).
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, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat".
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 ]
Its defining ideal is the principal ideal of k[x] generated by the initial term of f, namely y 2 − x 2 = 0. The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x.