Abstract
This paper explores proofs of the isoperimetric inequality for 4-connected shapes on the integer grid ℤ2, and its geometric meaning. Pictorially, we discuss ways to place a maximal number unit square tiles on a chess board so that the shape they form has a minimal number of unit square neighbors. Previous works have shown that “digital spheres” have a minimum of neighbors for their area. We here characterize all shapes that are optimal and show that they are all close to being digital spheres. In addition, we show a similar result when the 8-connectivity metric is assumed (i.e. connectivity through vertices or edges, instead of edge connectivity as in 4-connectivity).
This research was supported in part by the Israeli Ministry of Science Infrastructural Grant No. 3-942 and in part by the Devorah fund.
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Altshuler, Y., Yanovsky, V., Vainsencher, D., Wagner, I.A., Bruckstein, A.M. (2006). On Minimal Perimeter Polyminoes. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds) Discrete Geometry for Computer Imagery. DGCI 2006. Lecture Notes in Computer Science, vol 4245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11907350_2
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DOI: https://doi.org/10.1007/11907350_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-47651-1
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