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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.
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.
A variety is called convex if the pullback of the tangent bundle to a stable rational curve: has globally generated sections. [2] Geometrically this implies the curve is free to move around X {\displaystyle X} infinitesimally without any obstruction.
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
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]
A subset C of S is convex if, for all x and y in C, the line segment connecting x and y is included in C. This means that the affine combination (1 − t)x + ty belongs to C for all x,y in C and t in the interval [0, 1]. This implies that convexity is invariant under affine transformations.
Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are ...