Convexity in Elementary Calculus: Some Geometric Equivalences

Abstract

What is convexity? For the beginning student of calculus, it is a concept usually introduced to explain the geometric significance of the second derivative. It is easy enough to illustrate graphically or describe verbally (holds water), but what sort of definition is appropriate in this setting? There are several equivalent ways to formulate the definition and relate it to the sign of the second derivative, but many widely used textbooks do not reveal the variety of alternatives or, in the case of some more advanced ones, give proofs of equivalence which are unnecessarily technical or nonintuitive. For texts which contain a more thorough discussion of convexity, see [1], [2]. Here we will look at six well-known formulations of convexity which have easily visualized geometric interpretations and prove their equivalence in an elementary way. We hope that the presentation of some of this (admittedly optional) material in the classroom may hint at the fundamental character of convexity and help to dispel the possible suspicion that the second derivative test for convexity is little more than a disguised definition. Though infrequently seen in calculus texts, perhaps the most common definition of convexity for an arbitrary function is as follows: