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.

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

<math>f(tx+(1-t)y)\leq t f(x)+(1-t)f(y).</math>

A function is also said to be strictly convex if

<math>f(tx+(1-t)y) < t f(x)+(1-t)f(y).</math>

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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