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It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex. The absolute value function () = | | is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point = It is not strictly convex.
The real absolute value function is a piecewise linear, convex function. For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself).
Rigorously, a subderivative of a convex function : at a point in the open interval is a real number such that () for all .By the converse of the mean value theorem, the set of subderivatives at for a convex function is a nonempty closed interval [,], where and are the one-sided limits = (), = + ().
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.
A linear functional on a topological vector space is continuous if and only if its absolute value | | is continuous, which happens if and only if there exists a continuous seminorm on such that | | on the domain of . [17] If is a locally convex space then this statement remains true when the linear functional is defined on a proper vector ...
In general, the value of the norm is dependent on the spectrum of : For a vector with a Euclidean norm of one, the value of ‖ ‖ is bounded from below and above by the smallest and largest absolute eigenvalues of respectively, where the bounds are achieved if coincides with the corresponding (normalized) eigenvectors.
The absolute value of a complex number is a positively homogeneous function of degree over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.
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.