Convex Function - Properties

Properties

Suppose f is a function of one real variable defined on an interval, and let

(note that is the slope of the purple line in the above drawing; note also that the function R is symmetric in ). f is convex if and only if is monotonically non-decreasing in, for fixed (or viceversa). This characterization of convexity is quite useful to prove the following results.

A convex function f defined on some open interval C is continuous on C and Lipschitz continuous on any closed subinterval. f admits left and right derivatives, and these are monotonically non-decreasing. As a consequence, f is differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C (an example is shown in the examples' section).

A function is midpoint convex on an interval C if

for all and in C. This condition is only slightly weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex by Sierpinski Theorem. In particular, a continuous function that is midpoint convex will be convex.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. For the basic case of a differentiable function from (a subset of) the real numbers to the real numbers, "convex" is equivalent to "increasing at an increasing rate".

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents:

for all x and y in the interval. In particular, if f '(c) = 0, then c is a global minimum of f(x).

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. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of f(x) = x4 is f "(x) = 12 x2, which is zero for x = 0, but x4 is strictly convex.

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

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 sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function.

Jensen's inequality applies to every convex function f. If X is a random variable taking values in the domain of f, then (Here denotes the mathematical expectation.)

If a function f is convex, and f(0) ≤ 0, then f is superadditive on the positive half-axis. Proof:

• since f is convex, let y = 0, for every
•