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In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class. The convexity of a measure μ on n-dimensional Euclidean space R n in the sense above is closely related to the convexity of its probability density function. [2]
This characterization of convexity is quite useful to prove the following results. A convex function of one real variable defined on some open interval is continuous on . admits left and right derivatives, and these are monotonically non-decreasing. In addition, the left derivative is left-continuous and the right-derivative is right-continuous.
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.
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces ¯, (,) in quantum cohomology. [ 1 ] : §1 [ 2 ] [ 3 ] These moduli spaces are smooth orbifolds whenever the target space is convex.
Convex conjugate - a dual of a real functional in a vector space. Can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. Convex curve - a plane curve that lies entirely on one side of each of its supporting lines. The interior of a closed convex curve is a convex set.
Two other theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. [ 2 ] The result is named for Constantin Carathéodory , who proved the theorem in 1911 for the case when P {\displaystyle P} is compact . [ 3 ]
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.