Optimal general Hermite-Hadamard-type inequalities in a ball and their applications in multidimensional numerical integration

In this paper, we are interested in the problem of approximation of a definite integral over a ball of a given function f in d-dimensional space when, rather than function evaluations, a number of integrals over certain (d−1)-dimensional hyperspheres are only available. In this context several families of ‘extended’ multidimensional integration formulas based on a weighted sum of integrals over some hyperspheres can be defined. The special cases include multivariate analogues of the well-known midpoint rule and the trapezoidal rule. Basic properties of these families are derived, in particular, we show that they all satisfy a multivariate version of Hermite–Hadamard inequality. As an immediate consequence of this inequality, we derive explicit expressions of the best constants, which appear in their optimal error estimates. Theoretical and numerical results show that the proposed method reaches at least the second order of approximation. We present several numerical examples to illustrate various features of these new cubature formulas.