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Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti

Published online by Cambridge University Press:  14 July 2016

B. Edwin Blaisdell  and
Herbert Solomon
B. Edwin Blaisdell*
Affiliation:
Linus Pauling Institute of Science and Medicine
Herbert Solomon*
Affiliation:
Stanford University
*
∗ Postal address: Linus Pauling Institute of Science and Medicine, 440 Page Mill Road, Palo Alto, CA 94306, U.S.A.
∗∗ Postal address: Department of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305, U.S.A.
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Abstract

A conjecture of Palásti [11] that the limiting packing density β d in a space of dimension d equals β d where ß is the limiting packing density in one dimension continues to be studied, but with inconsistent results. Some recent correspondence in this journal [7], [8], [13], [14], [15], [16], [18], [19], [20] as well as some other papers indicate a lively interest in the subject. In a prior study [3], we demonstrated that the conjectured value in two dimensions was smaller than the actual density. Here we demonstrate that this is also so in three and four dimensions and that the discrepancy increases with dimension.

Keywords

SEQUENTIAL RANDOM PACKINGPALÁSTI CONJECTUREGEOMETRICAL PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 2 , June 1982 , pp. 382 - 390
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Partial support by Office of Naval Research Contract No. N00014-76-C-0475, Stanford University.

References

[1] Akeda, Y. and Hori, M. (1975) Numerical test of Palásti's conjecture on two-dimensional random packing density. Nature, London 254, 318–319.Google Scholar
[2] Akeda, Y. and Hori, M. (1976) On random sequential packing in two and three dimensions. Biometrika 63, 361–366.Google Scholar
[3] Blaisdell, B. E. and Solomon, H. (1970) On random sequential packing in the plane and a conjecture of Palásti. J. Appl. Prob. 7, 667–698.Google Scholar
[4] Dixon, W. J. and Brown, M. B., (Eds.) (1977) BMDP-77. University of California Press, Berkeley.Google Scholar
[5] Dvoretsky, A. and Robbins, H. (1964) On the ‘parking’ problem. Publ. Math. Inst. Hung. Acad. Sci. 9, 209–224.Google Scholar
[6] Finegold, L. and Donnell, J. T. (1979) Maximum density of random placing of membrane particles. Nature, London 278, 443–445.Google Scholar
[7] Hori, M. (1979) On Weiner's proof of the Palásti conjecture. J. Appl. Prob. 16, 702–706.Google Scholar
[8] Hori, M. (1980) Comments on the second letter of Weiner. J. Appl. Prob. 17, 888–889.CrossRefGoogle Scholar
[9] Jodrey, W. S. and Tory, E. M. (1980) Random sequential packing in Rn. J. Statist. Comput. Simul. 10, 87–93.CrossRefGoogle Scholar
[10] Mackenzie, J. K. (1962) Sequential filling of a line by intervals placed at random and its application to linear adsorption. J. Chem. Phys. 37, 723–728.Google Scholar
[11] Palásti, I. (1960) On some random space filling problems. Publ. Math. Inst. Hung. Acad. Sci. 5, 353–359.Google Scholar
[12] Renyi, A. (1958) On a one dimensional problem concerning space filling. Publ. Math. Inst. Hung. Acad. Sci. 3, 109–127.Google Scholar
[13] Tanemura, M. (1979) Has the Palásti conjecture been proved?: A criticism of a paper by H. J. Weiner. J. Appl. Prob. 16, 697–698.Google Scholar
[14] Tanemura, M. (1980) Some comments on the letters by H. J. Weiner, J. Appl. Prob. 17, 884–447.Google Scholar
[15] Tory, E. M. and Pickard, D. K. (1979) Some comments on ‘Sequential random packing in the plane by H. J. Weiner’. J. Appl. Prob. 16, 699–702.Google Scholar
[16] Tory, E. M. and Pickard, D. K. (1980) A critique of Weiner's work on Palásti's conjecture. J. Appl. Prob. 17, 880–884.Google Scholar
[17] Weiner, H. J. (1978) Sequential random packing in the plane. J. Appl. Prob. 15, 803–814.Google Scholar
[18] Weiner, H. J. (1979) Reply to letters of M. Tanemura and E. M. Tory and D. K. Pickard. J. Appl. Prob. 16, 706–707.Google Scholar
[19] Weiner, H. J. (1980) Further comments on a paper by H. J. Weiner. J. Appl. Prob. 17, 878–880.Google Scholar
[20] Weiner, H. J. (1980) Reply to remarks of Professor M. Hori. J. Appl. Prob. 17, 890–892.Google Scholar