Abstract
We discuss packings of sequences of convex bodies of Euclideann-spaceE n in a box and particularly in a cube. Following an Auerbach-Banach-Mazur-Ulam problem from the well-knownScottish Book, results of this kind are called potato-sack theorems. We consider on-line packing methods which work under the restriction that during the packing process we are given each succeeding “potato” only when the preceding one has been packed. One of our on-line methods enables us to pack into the cube of sided>1 inE n every sequence of convex bodies of diameters at most 1 whose total volume does not exceed (\((d - 1)(\sqrt d - 1)^{2(n - 1)} /n!.\)). Asymptotically, asd→∞, this volume is as good as that given by the non-on-line methods previously known.
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This research was done during the academic year 1987/88, while the first author was visiting the City College of the City University of New York. The second author was supported in part by Office of Naval Research Grant N00014-85-K-0147.
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Lassak, M., Zhang, J. An on-line potato-sack theorem. Discrete Comput Geom 6, 1–7 (1991). https://doi.org/10.1007/BF02574670
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DOI: https://doi.org/10.1007/BF02574670