High-Dimensional Cubes, Balls and Spherically Symmetric Distributions

The main objective of this chapter is development of accurate approximations for the volume of intersection of a cube and a ball in \(\mathbb {R}^d\) . In Chap. 2 , the approximations developed in this chapter will be the cornerstone of more evolved approximations and methods of construction of efficient exploration strategies in global optimization and space-filling designs in high-dimensional sets.