Convex

An object is said to be convex if for any pair of points within the object, any point on the line that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.

Table of contents

1 Convex set 2 Convex function 2.1 Properties of convex functions 2.2 Examples of convex functions

Convex set

In mathematics, convexity can be defined for subsets of any real or complex vector space. Such a subset C is said to be convex if, for all x and y in C and all t in the interval [0,1], the point tx + (1-t)y is in C. In words, every point on the straight line connecting x and y is in C

The convex subsets of R (the set of real numbers) are simply the intervals of R. Some examples of convex subsets of Euclidean 3-space are the Archimedean solids and the Platonic solids. The Kepler solids are examples of non-convex sets.

The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice. This also means that any subset A of the vector space is contained within a smallest convex set (called the convex hull of A), namely the intersection of all convex sets containing A.

One application of convex hulls is found in efficiency frontier analysis. Efficiency is assumed to be a monotonic function of each of finitely many of real variables. Each one of finitely many data points is in exactly one hull, and is considered more efficient than all data points in hulls contained within its own hull. A particle whose velocity vector has a value of a for all coordinates representing maximized variables, and a value less than a for all minimized variables, will pass through the hulls in increasing order of efficiency.

Convex function

A real-valued function f defined on an interval (or on any convex subset of some vector space) is called convex if for any two points x and y in its domain and any t in [0,1], we have

A function is also said to be strictly convex if

One may compare this definition of convexity and that for sets, and note that a function is convex if, and only if, the region of the plane lying above the graph of said function is a convex set.

Properties of convex functions

A convex function defined on some open set is continuous on the whole interval and differentiable at all but at most countably many points. A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there and strictly convex if and only if its second derivative is positive; this gives a practical test for convexity.

Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.

Examples of convex functions

• The second derivative of x² is 2; it follows that x² is a convex function of x.
• The absolute value function |x| is convex, even though it does not have a derivative at x=0.
• The function f(x)=x is convex but not strictly convex.
• The function x³ has second derivative 6x; thus it is convex for x ≥ 0 and concave for x ≤ 0.