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The function () = has ″ = >, so f is a convex function. 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.
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.
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
This generalizes the idea of convexity in Euclidean geometry, where given two points , in a convex set , all of the points + are contained in that set. There is a vector field X U p {\displaystyle {\mathcal {X}}_{U_{p}}} in a neighborhood U p {\displaystyle U_{p}} of p {\displaystyle p} transporting p {\displaystyle p} to each point p ′ ∈ ...
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 .
A plane curve is called convex if it has a supporting line through each of its points. [8] [9] For example, the graph of a convex function has a supporting line below the graph through each of its points. More strongly, at the points where the function has a derivative, there is exactly one supporting line, the tangent line. [10]
The two distinguished points are examples of extreme points of a convex set that are not exposed In mathematics, an exposed point of a convex set C {\displaystyle C} is a point x ∈ C {\displaystyle x\in C} at which some continuous linear functional attains its strict maximum over C {\displaystyle C} . [ 1 ]
The convex hull of the extreme points of forms a convex subset of so the main burden of the proof is to show that there are enough extreme points so that their convex hull covers all of . For this reason, the following corollary to the above theorem is also often called the Krein–Milman theorem.