The Equivalence of Chebyshev’s Inequality to the Hermite-hadamard Inequality

equality holds in either side only for the affine functions (i.e., for the functions of the form mx+ n). The middle point (a + b)/2 represents the barycenter of the probability measure 1 b−adx (viewed as a mass distribution over the interval [a, b]), while a and b represent the extreme points of [a, b]. Thus the Hermite-Hadamard inequality could be seen as a precursor of Choquet’s theory. See [8] for details and further comments. The optimal transport theory offers more insights into the mechanism of this inequality. In fact, from the point of view of that theory, the barycenter