Interval Methods for Verifying Structural Optimality of Circle Packing Configurations in the Unit Square

The paper is dealing with the problem of finding the densest packings of equal circles in the unit square. Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem. With this method it became possible to solve the previously open problems of packing 28, 29, and 30 circles in the numerical sense: tight guaranteed enclosures were given for all the optimal solutions and for the optimum value. The present paper completes the optimality proofs for these cases by determining all the optimal solutions in the geometric sense. Namely, it is proved that the currently best-known packing structures result in optimal packings, and moreover, apart from symmetric configurations and the movement of well-identified free circles, these are the only optimal packings. The required statements are verified with mathematical rigor using interval arithmetic tools.