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A Single Beta-Complex Solves All Geometry Problems in a Molecule

Geometry-related problems in bio-molecules are new challenges as well as opportunities for geometers since it is now strongly agreed that the shape of molecules mostly determines their functions. To analyze the shape of molecule, a systematic approach of handling the proximity among constituting atoms is inevitable. The Voronoi diagram of atoms and its derivative structures have proven their powerful capabilities on analyzing the structure problems of bio-molecules related with geometry or shape of molecules. In this paper, we show that a single ¯-complex of a molecule can be used to efficiently solve many bio-molecular problems which are based on some geometry among atoms in the molecule. We explain the ¯-complex in the context of molecular geometry and shape analysis based on the Voronoi diagram of atoms and the quasi-triangulation. We also compare the capabilities of ¯-complex with the (weighted) ®-complex and (weighted) ®-shape which are based on the power diagram.