Existence, uniqueness, and algorithmic computation of general lilypond systems

摘要

The lilypond system based on a locally finite subset phi of the Euclidean space R-n is defined as follows. At time 0 every point of phi starts growing with unit speed in all directions to form a system of balls in which any particular ball ceases its growth at the instant that it collides with another ball. Based on a more formal definition of lilypond systems given in [1], we will prove that these systems exist and are uniquely determined. Our approach applies to the far more general setting, where p is a locally finite subset of some space X equipped with a pseudo-metric d. We will also derive an algorithm approximating the system with at least linearly decreasing error. Several examples will illustrate our general results. (c) 2005 Wiley Periodicals, Inc.