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Convex Function

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ConvexFunction

A convex function on an interval I subset= R is a real-valued function f:I->R satisfying

f[lambdax_1+(1-lambda)x_2]<=lambdaf(x_1)+(1-lambda)f(x_2)

for x_1,x_2 in I and 0<=lambda<=1 (Boyd and Vandenberghe 2004, p. 67). For a continuous function, it suffices to check the midpoint condition f((x_1+x_2)/2)<=(f(x_1)+f(x_2))/2 for every pair of points in the interval (Rudin 1976, p. 101; cf. Gradshteyn and Ryzhik 2000, p. 1132).

If f is twice differentiable on an open interval I, then a necessary and sufficient condition for convexity is f^('')(x)>=0 for all x in I (Boyd and Vandenberghe 2004, p. 71).

If the defining inequality is strict whenever x_1!=x_2 and 0<lambda<1, then f is called strictly convex (Boyd and Vandenberghe 2004, p. 67).

A convex function is continuous on the interior of its interval, but continuity at an included endpoint need not hold (Boyd and Vandenberghe 2004, p. 68). For example, the function on [0,1] given by f(0)=1 and f(x)=0 for 0<x<=1 is convex and discontinuous at 0.

Examples of convex functions include x^p for p=1 or even p>=2, xlnx for x>0, and |x| for all x. If the sign of the inequality is reversed, the function is called concave.

See also

Concave Function, Convex, Interval, Logarithmically Convex Function, McLeod Conjecture, Subadditive Function

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References

Boyd, S. and Vandenberghe, L. Convex Optimization. Cambridge, England: Cambridge University Press, 2004. https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf.Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How Likely Is a Function to Be Convex?" Math. Mag. 61, 211-219, 1988.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1132, 2000.Rudin, W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976.Webster, R. Convexity. Oxford, England: Oxford University Press, 1995.

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Convex Function

Cite this as:

Weisstein, Eric W. "Convex Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConvexFunction.html

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