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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. When X is uniformly convex, it admits an equivalent norm with power type modulus of ...
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:
is a convex set. [2] The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
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.
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 ...