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

I.e., a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. A function is also said to be strictly convex if

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

for any t in (0,1).

Logarithmically convex function

A function such that f(x) > 0 for all x is said to be logarithmically convex function if logf(x) is a convex function of x.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example f(x) = x² is a convex function, but logf(x) = logx² = 2logx is not a convex function and thus f(x) = x² is not logarithmically convex. On the other hand is logarithmically convex since is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals. (See also Bohr-Mollerup_theorem.)

The definition is easily extended to functions where still we have f > 0, in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals [a,b].

Properties of convex functions

A convex function f defined on some convex open interval C is continuous on the whole C and differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the border.

A continuous function on C is convex if and only if

for any x and y in C.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval.

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: f(y) ≥ f(x) + f'(x) (y - x) for all x and y in the interval.

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the opposite is not true, as shown by f(x) = x⁴.

More generally, a continuous, twice differentiable function of multiple variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.

If two functions f and g are convex, then so is any weighted combination a f + b g with non-negative coefficients a and b.

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

For a convex function f, the level sets {x|f(x)<a} and {x|f(x)≤a} with a∈R are convex sets.

A convex function respects Jensen's inequality.

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.

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