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We present filling as a new type of spatial subdivision problem that isrelated to covering and packing. Filling addresses the optimal placement ofoverlapping objects lying entirely inside an arbitrary shape so as to cover themost interior volume. In n-dimensional space, if the objects are polydispersen-balls, we show that solutions correspond to sets of maximal n-balls and thesolution space can reduced to the medial axis of a shape. We examine thestructure of the solution space in two dimensions. For the filling of polygons,we provide detailed descriptions of a heuristic and a genetic algorithm forfinding solutions of maximal discs. We also consider the properties of idealdistributions of N discs in polygons as N approaches infinity.

Optimal Fillings - A new spatial subdivision problem related to packing and covering