Convex set
An object is convex if any point lying directly between two points of the object is also in the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.
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.
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.
A real-valued function f defined on an interval (or on any convex set) 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) ≤ 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.