Jensen’s Inequality as a Criterion for the Convexity of a Continuous Function

Many inequalities in mathematics can be derived from a few basic inequalities. One such inequalities is the classical integral inequality of Jensen, which is used to obtain various estimates of integrals of functions belonging to different c lasses. F or e xample, a s pecial c ase o f t he i ntegral H older ¨ inequality for functions of class L p , p ≥ 1, can be obtained as a consequence of Jensen’s inequality, from which the embedding of spaces L q ⊂ L p , 1 ≤ q ≤ p can be easily derived. In terms of probability measure, Jensen’s inequality states that the value of a convex function at the expectation point of a vector random variable ξ does not exceed the expectation of this convex function of the random variable ξ: f(Eξ) ≤ Ef(ξ). In this paper we prove that if Jensen’s inequality holds for a continuous function f for any simplex, then f is convex downwards. We introduce the concept of the convexity defect of a continuous function, so that it is negative for convex functions. Using the linear properties of the convexity defect, we prove an integral criterion for δ-convexity of a continuous function. It states that if the convexity defect does not exceed a quadratic function of the diameter of the simplex, then the function is δ-convex. From this criterion we obtain an integral condition for twice almost everywhere differentiability o f a continuous function.