The Hermite-Hadamard inequality for convex functions on a global NPC space
``According to the classical Hermite-Hadamard inequality, the mean value of a continuous convex function \(f: [a,b]\to \mathbb{R}\) lies between the value of \(f\) at the midpoint of the interval \([a,b]\) and the arithmetic mean of the values of \(f\) at the endpoints of this interval, that is \[ f\Biggl({a+b\over 2}\Biggr)\leq {1\over b-a} \int^b_a f(x)\,dx\leq {f(a)+ f(b)\over 2}.\tag{HH} \] Choquet's theory offers a considerable insight into this answer, based on the fact that \((a+b)/2\) is the barycenter of the interval \([a,b]\) (with respect to the uniform distribution of mass \({dx\over b-a}\)), and the right-hand side of (HH) represents the mean value of \(f\) with respect to the probability measure \(\lambda={1\over 2}\partial_a+{1\over 2}\partial_b\), supported on the extreme points of the interval \([a,b]\).'' \(\dots\) ``We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature.'' \dots ``A globl NPC space is a complete metric space \(M= (M,d)\) for which the following inequality holds true: for each pair of points \(x_0,x_1\in M\) there exists a point \(y\in M\) such that, for all points \(z\in E\), \[ d^2(z,y)\leq \tfrac12 d^2(z,x_0)+\tfrac12 d^2(z, x_1)-\tfrac14 d^2(x_0, x_1).\text{ ''} \]
scientific article; zbMATH DE number 5563869