Abstract
We adapt a construction of Klee (1981) to find a packing of unit balls in ℓ p (1≤p<∞) which is efficient in the sense that enlarging the radius of each ball to any R>21−1/p covers the whole space. We show that the value 21−1/p is optimal.
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Ayerbe Toledano, J.M., Domínguez Benavides, T., López Acedo, G.: Measures of Non-Compactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997)
Böröczky, K.: Closest packing and loosest covering of the space with balls. Stud. Sci. Math. Hung. 21, 79–89 (1986)
Burlak, J.A.C., Rankin, R.A., Robertson, A.P.: The packing of spheres in the space ℓ p . Proc. Glasg. Math. Assoc. 4, 22–25 (1958)
Casini, E., Papini, P.L., Zanco, C.: Separation and Approximation in Normed Linear Spaces. International Series of Numerical Mathematics, vol. 76. Birkhäuser, Basel (1986)
Delone, B.N.: Geometry of positive quadratic forms. Usp. Mat. Nauk 3, 16–62 (1937)
Corson, H.H.: Collections of convex sets which cover a Banach space. Fund. Math. 49, 143–145 (1961)
Doyle, P.G., Lagarias, J.C., Randall, D.: Self-packing of centrally symmetric convex bodies in R 2. Discrete Comput. Geom. 8, 171–189 (1992)
Fejes Tóth, L.: Close packing and loose covering with balls. Publ. Math. Debr. 23, 323–326 (1976)
Henk, M.: Free planes in lattice sphere packings. Adv. Geom. 5, 137–144 (2005)
Klee, V.: Dispersed Chebyshev sets and coverings by balls. Math. Ann. 257, 251–260 (1981)
Kottman, C.A.: Packing and reflexivity in Banach spaces. Trans. Am. Math. Soc. 150, 565–576 (1970)
Kryczka, A., Prus, S.: Separated sequences in nonreflexive Banach spaces. Proc. Am. Math. Soc. 129, 155–163 (2000)
Linhart, J.: Closest packings and closest coverings by translates of a convex disc. Stud. Sci. Math. Hung. 13, 157–162 (1978)
Rogers, C.A.: A note on coverings and packings. J. Lond. Math. Soc. 25, 327–331 (1950)
Rogers, C.A.: Lattices in Banach spaces. Mitt. Math. Sem. Giessen 165, 155–167 (1984)
Ryškov, S.S.: Density of an (r,R)-system. Mat. Zametki 16, 447–454 (1974)
Schürmann, A., Vallentin, F.: Computational approaches to lattice packing and covering problems. Discrete Comput. Geom. 35, 73–116 (2006)
Wells, J.H., Williams, L.R.: Embeddings and Extensions in Analysis. Springer, New York (1975)
Zong, C.: From deep holes to free planes. Bull. Am. Math. Soc. (N.S.) 39, 533–555 (2002)
Zong, C.: Simultaneous packing and covering in the Euclidean plane. Monatsh. Math. 134, 247–255 (2002)
Zong, C.: Simultaneous packing and covering of centrally symmetric convex bodies. In: IV International Conference in Stochastic Geometry, Convex Bodies, Empirical Measures & Applications to Engineering Science, vol. II, Tropea, 2001. Rend. Circ. Mat. Palermo (2) Suppl. 70, part II, 387–396 (2002)
Zong, C.: Simultaneous packing and covering in three-dimensional Euclidean space. J. Lond. Math. Soc. (2) 67, 29–40 (2003)
Zong, C.: The simultaneous packing and covering constants in the plane. Adv. Math. 218, 653–672 (2008)
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Dedicated to the memory of Victor Klee.
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Swanepoel, K.J. Simultaneous Packing and Covering in Sequence Spaces. Discrete Comput Geom 42, 335–340 (2009). https://doi.org/10.1007/s00454-009-9189-8
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DOI: https://doi.org/10.1007/s00454-009-9189-8