Strongly normal sets of tiles in N dimensions

The first and third authors and others [2,8,9,10,11,12] have studied sets of “tiles” (a generalization of pixels or voxels) in two and three dimensions that have a property called strong normality (SN): For any tile P , only finitely many tiles intersect P , and any nonempty intersection of these tiles must also intersect P . This paper presents extensions of the basic results about SN sets of tiles to n dimensions. One of our results is that if SN holds for every n+1 or fewer tiles in a locally finite set of tiles in R, then the entire set of tiles is SN. Other results are that SN is equivalent to hereditary local contractibility, that simpleness of a tile in an SN set of tiles is equivalent to contractibility of its shared subset, and that deletion of a simple tile in an SN set of tiles preserves the homotopy type of the union of all the tiles. 1 Email: [email protected] 2 Email: [email protected] 3 Present address: Computer Science Dept., Queens College, Flushing, NY 11367, U.S.A. 4 Email: [email protected] c ©2001 Published by Elsevier Science B. V.