# Minimal Surface Convex Hulls of Spheres

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## Abstract

We present and solve a new computational geometry optimization problem. Spheres with given radii should be arranged such that (a) they do not overlap and (b) the surface area of the boundary of the convex hull enclosing the spheres is minimized. An additional constraint could be to fit the spheres into a specified geometry, e.g., a rectangular solid. To tackle the problem, we derive closed non-convex NLP models for this sphere arrangement or sphere packing problem. For two spheres, we prove that the minimal area of the boundary of the convex hull is identical to the sum of the surface areas of the two spheres. For special configurations of spheres we provide theoretical insights and we compute analytically minimal-area configurations. Numerically, we have solved problems containing up to 200 spheres.

## Keywords

Packing problem Convex hull minimization Isoperimetric inequality Computational geometry Non-convex nonlinear programming Global optimization## Mathematics Subject Classification (2010)

51 90## Notes

### Acknowledgements

We thank Julius Näumann (Student, TU Darmstadt, Darmstadt, Germany) for producing the graphics software for this paper. Thanks are directed to Dr. Jens Schulz (Lufthansa Systems GmbH, Berlin, Germany), Prof. Dr. Julia Kallrath (Hochschule Darmstadt, Darmstadt, Germany), Julius Näumann, Dr. Fritz Näumann (Consultant, Weisenheim am Berg, Germany), and Dr. Wolfgang Heinecke (Edenkoben, Germany) for their careful reading of and feedback on the manuscript.

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