Convex Function - Examples

Examples

• The function has at all points, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
• The function has, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.
• The absolute value function is convex, even though it does not have a derivative at the point x = 0. It is not strictly convex.
• The function for 1 ≤ p is convex.
• The exponential function is convex. It is also strictly convex, since, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function is logarithmically convex if f is a convex function. The term "superconvex" is sometimes used instead.
• The function f with domain defined by f(0) = f(1) = 1, f(x) = 0 for 0 < x < 1 is convex; it is continuous on the open interval (0, 1), but not continuous at 0 and 1.
• The function x3 has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0.
• Every linear transformation taking values in is convex but not strictly convex, since if f is linear, then This statement also holds if we replace "convex" by "concave".
• Every affine function taking values in, i.e., each function of the form, is simultaneously convex and concave.
• Every norm is a convex function, by the triangle inequality and positive homogeneity.
• Examples of functions that are monotonically increasing but not convex include and g(x) = log(x).
• Examples of functions that are convex but not monotonically increasing include and .
• The function f(x) = 1/x has which is greater than 0 if x > 0, so f(x) is convex on the interval (0, +∞). It is concave on the interval (-∞,0).
• The function f(x) = 1/x2, with f(0) = +∞, is convex on the interval (0, +∞) and convex on the interval (-∞,0), but not convex on the interval (-∞, +∞), because of the singularity at x = 0.