A Contribution to the Development of Functional Thinking Related to Convexity

Abstract

When a liquid (water) flows into a vessel at the constant inflow rate, then the height filling function is convex or concave depending on the way how the level of the liquid changes. When the level changes accelerating or slowing down, the function is convex or concave, respectively. This vivid interpretation holds in general, namely we prove that given a strictly increasing convex (concave) continuous function, then there exists a vessel such that its height filling function is equal to the given function. (A fact that seems to be new.) We also hope that our paper could exemplify the case of a research project to be assigned to excellent students.