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Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. [11] A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function f {\displaystyle f} is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
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. When X is uniformly convex, it admits an equivalent norm with power type modulus of ...
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]
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets , often with applications in convex minimization , a subdomain of optimization theory .
This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. [1] Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x ...
Every convex function is pseudoconvex, but the converse is not true. For example, the function f ( x ) = x + x 3 {\displaystyle f(x)=x+x^{3}} is pseudoconvex but not convex. Similarly, any pseudoconvex function is quasiconvex ; but the converse is not true, since the function f ( x ) = x 3 {\displaystyle f(x)=x^{3}} is quasiconvex but not ...
Strictly logarithmically convex if is strictly convex. Here we interpret as . Explicitly, f is logarithmically convex if and only if, for all x 1, x 2 ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold:
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.