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then is called strictly convex. [1]Convex functions are related to convex sets. Specifically, the function is convex if and only if its epigraph. A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set.
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.
A plane curve is the image of any continuous function from an interval to the Euclidean plane.Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivative
This is a generalization of the concept of strongly convex function; by taking () = we recover the definition of strong convexity. It is worth noting that some authors require the modulus ϕ {\displaystyle \phi } to be an increasing function, [ 17 ] but this condition is not required by all authors.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
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 .
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 bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex. [1] [2] [3] [4]