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# Convex

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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.

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

$f(tx+(1-t)y)\leq t f(x)+(1-t)f(y).$

A function is also said to be strictly convex if

$f(tx+(1-t)y) < t f(x)+(1-t)f(y).$

A convex function defined on some interval is continuous on the whole interval and differentiable at all but at most countably many points. A twice differentiable function is convex on an interval if and only if its second derivative is non-negative there.

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.

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