proof of criterion for convexity

Theorem 1.

Suppose f:(a,b)R is continuousMathworldPlanetmathPlanetmath and that, for all x,y(a,b),


Then f is convex.


We begin by showing that, for any natural numbersMathworldPlanetmath n and m2n,


by inductionMathworldPlanetmath. When n=1, there are three possibilities: m=1, m=0, and m=2. The first possibility is a hypothesisMathworldPlanetmathPlanetmath of the theoremMathworldPlanetmath being proven and the other two possibilities are trivial.

Assume that


for some n and all m2n. Let m be a number less than or equal to 2n+1. Then either m2n or 2n+1-m2n. In the former case we have

f(12mx+(2n-m)y2n+y2) 12(f(mx+(2n-m)y2n)+f(y))

In the other case, we can reverse the roles of x and y.

Now, every real number s has a binary expansion; in other words, there exists a sequence {mn} of integers such that limnmn/2n=s. If 0s1, then we also have 0mn2n so, by what we proved above,


Since f is assumed to be continuous, we may take the limit of both sides and conclude


which implies that f is convex. ∎

Title proof of criterion for convexity
Canonical name ProofOfCriterionForConvexity
Date of creation 2013-03-22 17:00:23
Last modified on 2013-03-22 17:00:23
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Proof
Classification msc 52A41
Classification msc 26A51
Classification msc 26B25