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Journal articles on the topic 'Packing dimension'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

STÄGER, D. V., and H. J. HERRMANN. "CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS." Fractals 26, no. 01 (February 2018): 1850013. http://dx.doi.org/10.1142/s0218348x18500135.

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Any space-filling packing of spheres can be cut by a plane to obtain a space-filling packing of disks. Here, we deal with space-filling packings generated using inversive geometry leading to exactly self-similar fractal packings. First, we prove that cutting along a random hyperplane leads in general to a packing with a fractal dimension of the one of the uncut packing minus one. Second, we find special cuts which can be constructed themselves by inversive geometry. Such special cuts have specific fractal dimensions, which we demonstrate by cutting a three- and a four-dimensional packing. The increase in the number of found special cuts with respect to a cutoff parameter suggests the existence of infinitely many topologies with distinct fractal dimensions.

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2

Conidis, Chris J. "A real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one." Journal of Symbolic Logic 77, no. 2 (June 2012): 447–74. http://dx.doi.org/10.2178/jsl/1333566632.

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AbstractRecently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

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3

MATTILA, PERTTI, and R. DANIEL MAULDIN. "Measure and dimension functions: measurability and densities." Mathematical Proceedings of the Cambridge Philosophical Society 121, no. 1 (January 1997): 81–100. http://dx.doi.org/10.1017/s0305004196001089.

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During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

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4

Das, Manav. "Billingsley's packing dimension." Proceedings of the American Mathematical Society 136, no. 01 (January 1, 2008): 273–79. http://dx.doi.org/10.1090/s0002-9939-07-09069-7.

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5

FALCONER, K. J., and J. D. HOWROYD. "Packing dimensions of projections and dimension profiles." Mathematical Proceedings of the Cambridge Philosophical Society 121, no. 2 (March 1997): 269–86. http://dx.doi.org/10.1017/s0305004196001375.

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6

Myjak, Józef. "Some typical properties of dimensions of sets and measures." Abstract and Applied Analysis 2005, no. 3 (2005): 239–54. http://dx.doi.org/10.1155/aaa.2005.239.

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This paper contains a review of recent results concerning typical properties of dimensions of sets and dimensions of measures. In particular, we are interested in the Hausdorff dimension, box dimension, and packing dimension of sets and in the Hausdorff dimension, box dimension, correlation dimension, concentration dimension, and local dimension of measures.

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7

HOWROYD, J. D. "Box and packing dimensions of projections and dimension profiles." Mathematical Proceedings of the Cambridge Philosophical Society 130, no. 1 (January 2001): 135–60. http://dx.doi.org/10.1017/s0305004100004849.

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For E a subset of ℝn and s ∈ [0, n] we define upper and lower box dimension profiles, B-dimsE and B-dimsE respectively, that are closely related to the box dimensions of the orthogonal projections of E onto subspaces of ℝn. In particular, the projection of E onto almost all m-dimensional subspaces has upper box dimension B-dimmE and lower box dimension B-dimmE. By defining a packing type measure with respect to s-dimensional kernels we are able to establish the connection to an analogous packing dimension theory.

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8

FALCONER, K. J., and M. JÄRVENPÄÄ. "Packing dimensions of sections of sets." Mathematical Proceedings of the Cambridge Philosophical Society 125, no. 1 (January 1999): 89–104. http://dx.doi.org/10.1017/s0305004198002977.

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We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

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9

Kontorovich, Alex, and Kei Nakamura. "Geometry and arithmetic of crystallographic sphere packings." Proceedings of the National Academy of Sciences 116, no. 2 (December 26, 2018): 436–41. http://dx.doi.org/10.1073/pnas.1721104116.

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We introduce the notion of a “crystallographic sphere packing,” defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions.

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10

PÖTZELBERGER, KLAUS. "The quantization dimension of distributions." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 3 (November 2001): 507–19. http://dx.doi.org/10.1017/s0305004101005357.

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We show that the asymptotic behaviour of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension.

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11

Downey, Rod, and Keng Meng Ng. "Effective Packing Dimension and Traceability." Notre Dame Journal of Formal Logic 51, no. 2 (April 2010): 279–90. http://dx.doi.org/10.1215/00294527-2010-017.

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12

Bayart, Frédéric, and Yanick Heurteaux. "Multifractal phenomena and packing dimension." Revista Matemática Iberoamericana 35, no. 3 (April 1, 2019): 767–804. http://dx.doi.org/10.4171/rmi/1069.

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13

Bishop, Christopher J., and Yuval Peres. "Packing dimension and Cartesian products." Transactions of the American Mathematical Society 348, no. 11 (1996): 4433–45. http://dx.doi.org/10.1090/s0002-9947-96-01750-3.

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14

Biran, P. "Symplectic Packing in Dimension 4." Geometric And Functional Analysis 7, no. 3 (July 1, 1997): 420–37. http://dx.doi.org/10.1007/s000390050014.

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15

Xiao, Yimin. "Packing dimension, Hausdorff dimension and Cartesian product sets." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 3 (October 1996): 535–46. http://dx.doi.org/10.1017/s030500410007506x.

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AbstractWe show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.

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16

STALLARD, GWYNETH M. "DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS." Bulletin of the London Mathematical Society 33, no. 6 (November 2001): 689–94. http://dx.doi.org/10.1112/s0024609301008426.

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It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

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17

BARAM, REZA MAHMOODI, and HANS J. HERRMANN. "SELF-SIMILAR SPACE-FILLING PACKINGS IN THREE DIMENSIONS." Fractals 12, no. 03 (September 2004): 293–301. http://dx.doi.org/10.1142/s0218348x04002549.

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We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bichromatic packing known up to now.

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18

WANG, BIN, WENLONG JI, LEGUI ZHANG, and XUAN LI. "THE RELATIONSHIP BETWEEN FRACTAL DIMENSIONS OF BESICOVITCH FUNCTION AND THE ORDER OF HADAMARD FRACTIONAL INTEGRAL." Fractals 28, no. 07 (November 2020): 2050128. http://dx.doi.org/10.1142/s0218348x20501285.

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In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of [Formula: see text] have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, [Formula: see text]-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral and fractal dimensions of Besicovitch function has also been given.

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19

Falconer, K. J., and J. D. Howroyd. "Projection theorems for box and packing dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (February 1996): 287–95. http://dx.doi.org/10.1017/s0305004100074168.

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AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

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20

Junqueira, Leonardo, and Reinaldo Morabito. "On solving three-dimensional open-dimension rectangular packing problems." Engineering Optimization 49, no. 5 (August 2, 2016): 733–45. http://dx.doi.org/10.1080/0305215x.2016.1208010.

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21

Baek, H. K. "Packing dimension and measure of homogeneous Cantor sets." Bulletin of the Australian Mathematical Society 74, no. 3 (December 2006): 443–48. http://dx.doi.org/10.1017/s000497270004048x.

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For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

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22

Lutz, Neil. "Fractal Intersections and Products via Algorithmic Dimension." ACM Transactions on Computation Theory 13, no. 3 (September 30, 2021): 1–15. http://dx.doi.org/10.1145/3460948.

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Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

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23

KHOSHNEVISAN, DAVAR, and YIMIN XIAO. "Packing-dimension profiles and fractional Brownian motion." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 1 (July 2008): 205–13. http://dx.doi.org/10.1017/s0305004108001394.

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AbstractIn order to compute the packing dimension of orthogonal projections Falconer and Howroyd [3] have introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd [5] introduced alternate s-dimensional packing dimension profiles P-Dims by using Caratheodory-type packing measures, and proved, among many other things, that P-DimsE = DimsE for all integers s > 0 and all analytic sets E ⊆ RN.The aim of this paper is to prove that P-DimsE = DimsE for all real numbers s > 0 and analytic sets E ⊆ RN. This answers a question of Howroyd [5, p. 159]. Our proof hinges on establishing a new property of fractional Brownian motion.

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24

Khoshnevisan, D., R. L. Schilling, and Y. Xiao. "Packing dimension profiles and Lévy processes." Bulletin of the London Mathematical Society 44, no. 5 (April 6, 2012): 931–43. http://dx.doi.org/10.1112/blms/bds022.

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25

JÄRVENPÄÄ, MAARIT. "Packing dimension, intersection measures, and isometries." Mathematical Proceedings of the Cambridge Philosophical Society 122, no. 3 (November 1997): 483–90. http://dx.doi.org/10.1017/s0305004197001941.

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26

Beliaev, D., E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Smirnov, and V. Suomala. "Packing dimension of mean porous measures." Journal of the London Mathematical Society 80, no. 2 (August 14, 2009): 514–30. http://dx.doi.org/10.1112/jlms/jdp040.

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27

Rams, Michał. "Packing dimension estimation for exceptional parameters." Israel Journal of Mathematics 130, no. 1 (December 2002): 125–44. http://dx.doi.org/10.1007/bf02764074.

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28

Talagrand, Michel, and Yimin Xiao. "Fractional Brownian motion and packing dimension." Journal of Theoretical Probability 9, no. 3 (July 1996): 579–93. http://dx.doi.org/10.1007/bf02214076.

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29

O'Rourke, Joseph. "Computational Geometry Column 31." International Journal of Computational Geometry & Applications 07, no. 04 (August 1997): 379–82. http://dx.doi.org/10.1142/s0218195997000235.

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30

Itoh, Yoshiaki, and Herbert Solomon. "Random sequential coding by Hamming distance." Journal of Applied Probability 23, no. 3 (September 1986): 688–95. http://dx.doi.org/10.2307/3214007.

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Here we introduce two simple models: simple cubic random packing and random packing by Hamming distance. Consider the packing density γ d of dimension d by cubic random packing. From computer simulations up to dimension 11, γ d+1/γ d seems to approach 1. Also, we give simulation results for random packing by Hamming distance and discuss the behavior of packing density when dimensionality is increased. For the case of Hamming distances of 2 or 3, d–α fits the simulation results of packing density where α is an empirical constant. The variance of packing density is larger when k is even and smaller when k is odd, where k represents Hamming distance.

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31

Itoh, Yoshiaki, and Herbert Solomon. "Random sequential coding by Hamming distance." Journal of Applied Probability 23, no. 03 (September 1986): 688–95. http://dx.doi.org/10.1017/s0021900200111842.

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Here we introduce two simple models: simple cubic random packing and random packing by Hamming distance. Consider the packing density γ d of dimension d by cubic random packing. From computer simulations up to dimension 11, γ d +1/γ d seems to approach 1. Also, we give simulation results for random packing by Hamming distance and discuss the behavior of packing density when dimensionality is increased. For the case of Hamming distances of 2 or 3, d–α fits the simulation results of packing density where α is an empirical constant. The variance of packing density is larger when k is even and smaller when k is odd, where k represents Hamming distance.

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32

Lévay, Sára, David Fischer, Ralf Stannarius, Balázs Szabó, Tamás Börzsönyi, and János Török. "Frustrated packing in a granular system under geometrical confinement." Soft Matter 14, no. 3 (2018): 396–404. http://dx.doi.org/10.1039/c7sm01900a.

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In this paper we show experimentally and numerically that in 2 + ε dimensions, realized by a container which is in one dimension slightly wider than the spheres, the particles organize themselves in a triangular lattice, while touching either the front or rear side of the container.

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33

Howroyd, J. D. "On Hausdorff and packing dimension of product spaces." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 4 (May 1996): 715–27. http://dx.doi.org/10.1017/s0305004100074545.

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AbstractWe show that for arbitrary metric spaces X and Y the following dimension inequalities hold:where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

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34

Pidcock, Elna, and W. D. Sam Motherwell. "Parameterization of the close packing of molecules in the unit cell." Acta Crystallographica Section B Structural Science 60, no. 6 (November 11, 2004): 725–33. http://dx.doi.org/10.1107/s0108768104022128.

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The box model of crystal packing describes unit cells in terms of a limited number of arrangements, or packing patterns, of molecular building blocks. Cell dimensions have been shown to relate to molecular dimensions in a systematic way. The distributions of pattern coefficients (cell length/molecular dimension) for thousands of structures belonging to P21/c, P\bar 1, P212121, P21 and C2/c are presented and are shown to be entirely consistent with the box model of crystal packing. Contributions to the form of the histograms from molecular orientation and molecular overlap are discussed. Gaussian fitting of the histograms has led to the parameterization of close packing within the unit cell and it is shown that molecular crystal structures are very similar to one another at a fundamental level.

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35

Zähle, Martina. "Dimensions of measures under orthogonal projections." Advances in Applied Probability 28, no. 2 (June 1996): 344–45. http://dx.doi.org/10.2307/1428059.

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Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝn given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

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36

REEVE, HENRY W. J. "The packing spectrum for Birkhoff averages on a self-affine repeller." Ergodic Theory and Dynamical Systems 32, no. 4 (September 8, 2011): 1444–70. http://dx.doi.org/10.1017/s0143385711000368.

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AbstractWe consider the multifractal analysis of Birkhoff averages of continuous potentials on a self-affine Sierpiński sponge. In particular, we give a variational principle for the packing dimension of the level sets. Furthermore, we prove that the packing spectrum is concave and continuous. We give a sufficient condition for the packing spectrum to be real analytic, but show that for general Hölder continuous potentials, this need not be the case. We also give a precise criterion for when the packing spectrum attains the full packing dimension of the repeller. Again, we present an example showing that this is not always the case.

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37

Zähle, Martina. "Dimensions of measures under orthogonal projections." Advances in Applied Probability 28, no. 02 (June 1996): 344–45. http://dx.doi.org/10.1017/s0001867800048497.

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Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝ n given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

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38

Nielsen, Ole A. "The Hausdorff and Packing Dimensions of Some Sets Related to Sierpiński Carpets." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 1073–88. http://dx.doi.org/10.4153/cjm-1999-047-4.

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AbstractThe Sierpiński carpets first considered by C.McMullen and later studied by Y. Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i) calculates theHausdorff, box (or Minkowski), and packing dimensions of themodified Sierpiński carpets and (ii) shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.

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39

Attia, Najmeddine, and Bilel Selmi. "Relative multifractal box-dimensions." Filomat 33, no. 9 (2019): 2841–59. http://dx.doi.org/10.2298/fil1909841a.

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Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

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40

WANG, JUN, and KUI YAO. "DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION." Fractals 25, no. 01 (February 2017): 1730001. http://dx.doi.org/10.1142/s0218348x1730001x.

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In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

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41

Viazovska, Maryna. "The sphere packing problem in dimension 8." Annals of Mathematics 185, no. 3 (May 1, 2017): 991–1015. http://dx.doi.org/10.4007/annals.2017.185.3.7.

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42

Cohn, Henry, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska. "The sphere packing problem in dimension 24." Annals of Mathematics 185, no. 3 (May 1, 2017): 1017–33. http://dx.doi.org/10.4007/annals.2017.185.3.8.

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43

JÄRVENPÄÄ, MAARIT. "Concerning the packing dimension of intersection measures." Mathematical Proceedings of the Cambridge Philosophical Society 121, no. 2 (March 1997): 287–96. http://dx.doi.org/10.1017/s0305004196001417.

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44

Davis, I. Lee, and Roger G. Carter. "Random particle packing by reduced dimension algorithms." Journal of Applied Physics 67, no. 2 (January 15, 1990): 1022–29. http://dx.doi.org/10.1063/1.345785.

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45

Conidis, Chris J. "Effective packing dimension of $\Pi ^0_1$-classes." Proceedings of the American Mathematical Society 136, no. 10 (May 15, 2008): 3655–62. http://dx.doi.org/10.1090/s0002-9939-08-09335-0.

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46

Berlinkov, Artemi. "Exact packing dimension in random recursive constructions." Probability Theory and Related Fields 126, no. 4 (August 1, 2003): 477–96. http://dx.doi.org/10.1007/s00440-003-0281-3.

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47

Liu, Jia, Bo Tan, and Jun Wu. "Graphs of continuous functions and packing dimension." Journal of Mathematical Analysis and Applications 435, no. 2 (March 2016): 1099–106. http://dx.doi.org/10.1016/j.jmaa.2015.11.011.

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48

Markiewicz, Tomasz G. "An Energy Efficient QAM Modulation with Multidimensional Signal Constellation." International Journal of Electronics and Telecommunications 62, no. 2 (June 1, 2016): 159–65. http://dx.doi.org/10.1515/eletel-2016-0022.

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Abstract Packing constellations points in higher dimensions, the concept of multidimensional modulation exploits the idea drawn from geometry for searching dense sphere packings in a given dimension, utilising it to minimise the average energy of the underlying constellations. The following work analyses the impact of spherical shaping of the constellations bound instead of the traditional, hyper-cubical bound. Balanced constellation schemes are obtained with the N-dimensional simplex merging algorithm. The performance of constellations of dimensions 2, 4 and 6 is compared to the performance of QAM modulations of equivalent throughputs in the sense of bits transmitted per complex (two-dimensional) symbols. The considered constellations give an approximately 0:7 dB to 1 dB gain in terms of BER over a standard QAM modulation.

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49

ZÄHLE, M. "THE AVERAGE FRACTAL DIMENSION AND PROJECTIONS OF MEASURES AND SETS IN Rn." Fractals 03, no. 04 (December 1995): 747–54. http://dx.doi.org/10.1142/s0218348x95000667.

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In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.

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50

HENK, MARTIN, GÜNTER M. ZIEGLER, and CHUANMING ZONG. "ON FREE PLANES IN LATTICE BALL PACKINGS." Bulletin of the London Mathematical Society 34, no. 3 (May 2002): 284–90. http://dx.doi.org/10.1112/s0024609301008888.

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This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d-dimensional packing lattice of balls one can find a four- dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

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