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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 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 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 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 ]
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 ...
In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater. [1] Informally, Slater's condition states that the feasible region must have an interior point (see technical details below).
For instance, the traveling salesman problem, NP-hard for arbitrary sets of points in the plane, is trivial for points in convex position: the optimal tour is the convex hull. [3] Similarly, the minimum-weight triangulation of planar point sets is NP-hard for arbitrary point sets, [ 4 ] but solvable in polynomial time by dynamic programming for ...
The subspace A is called a deformation retract of X. A deformation retraction is a special case of a homotopy equivalence. A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space X would imply that X is path connected (and in fact that X is contractible).