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In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at ...
Strictly convex space, a normed vector space for which the closed unit ball is a strictly convex set Topics referred to by the same term This disambiguation page lists articles associated with the title Strictly convex .
The Banach space (X, ǁ ⋅ ǁ) is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2.
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .
In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points. [1] [2] Formal Definition
It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.
Here, a set is strictly convex if every point of its boundary is an extreme point of the set, the unique maximizer of some linear function. [23] As the boundaries of strictly convex sets, these are the curves that lie in convex position, meaning that none of their points can be a convex combination of any other subset of its points. [24]
A closed convex subset of a topological vector space is called strictly convex if every one of its (topological) boundary points is an extreme point. [6] The unit ball of any Hilbert space is a strictly convex set. [6]