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2 Convex function |

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 segment 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*.

If one restricts to closed convex sets, they can actually be characterised as the intersections of closed half-spaces, lying to one side of a hyperplane. From what has just been said, it is clear that such intersections are convex, and they will also be closed sets. For the converse, one need the *supporting hyperplane theorem* in the form that for a given closed convex set C and point P outside it, there is a closed half-space H that contains C and not P. (That theorem is a case of the Hahn-Banach theorem of functional analysis, but less deep.)

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 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.

A convex function respects the Jensen's inequality.

- The second derivative of
*x*^{2}is*2*; it follows that*x*^{2}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
*x*^{3}has second derivative 6*x*; thus it is convex for*x*≥ 0 and concave for*x*≤ 0.