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A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex. In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or ...
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
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 .
A log-concave function is also quasi-concave. 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.
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.
In this graph, triangle 1-2-5 is convex, but path 2-3-4 is not, because it does not include one of the two shortest paths from 2 to 4. In metric graph theory , a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices.
If is strictly quasiconcave in for each and is convex-valued, then is single-valued, and thus is a continuous function rather than a correspondence. [ 15 ] If f {\displaystyle f} is concave in X × Θ {\displaystyle X\times \Theta } and C {\displaystyle C} has a convex graph, then f ∗ {\displaystyle f^{*}} is concave and C ∗ {\displaystyle ...
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.