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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.
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.
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 ′ ∈ ...
If is a real or complex vector space and if is the set of all seminorms on then the locally convex TVS topology, denoted by , that induces on is called the finest locally convex topology on . [37] This topology may also be described as the TVS-topology on having as a neighborhood base at the origin the set of all absorbing disks in . [37] Any ...
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 ]
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
The convex layers of a point set and their intersection with a halfplane. In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their vertices. The outermost one is the convex hull of the points and the rest are formed in the same way recursively.