Maximizing the sum of radii of balls inscribed in a polyhedral set

The sphere packing problem is one of the most applicable areas in mathematics which finds numerous applications in science and technology [1–4; 8; 9; 11–14]. We consider a maximization problem of a sum of radii of non-overlapping balls inscribed in a polyhedral set in Hilbert space. This problem is often formulated as the sphere packing problem. We extend the problem in Hilbert space as an optimal control problem with the terminal functional and constraints for the final moment. This problem belongs to a class of nonconvex optimal control problem and application of gradient methods does not always guarantee finding a global solution to the problem. We show that the problem in a finite dimensional case for three balls (spheres) is connected to well known Malfatti’s problem [16]. Malfatti’s generalized problem was examined in [6; 7] as the convex maximization problem employing the global optimality conditions of Strekalovsky [17].