By Larry Smith
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Additional info for Algebraic topology: Proc. conf. Goettingen 1984
Soc. Math. France 178 (1950) 152–161. -H. Kang: Covering Euclidean n-space by [F¨ uK05] Z. Fu translates of a convex body, manuscript. [Gr85] P. Gritzmann: Lattice covering of space with symmetric convex bodies, Mathematika 32 (1985) 311–315. [He98] A. Heppes: Research problem, Period. Math. Hungar. 36 (1998) 181–182. [Is98] D. Ismailescu: Covering the plane with copies of a convex disk, Discrete Comput. Geom. 20 (1998) 251–263. B. Kershner: The number of circles covering a set, Amer. J. Math. 61 (1939) 665–671.
In the latter case, the centers of the spherical balls form the vertex set of a regular four-dimensional crosspolytope [DaL*00]. Problem 16 (Larman, Zong) Consider a covering of Sd with n = 2d + 2 equal spherical balls of minimum radius, d > 3. Do the centers of the balls form the vertex set of a regular ddimensional crosspolytope? Actually, it is possible that for every d ≥ 3 and d + 2 ≤ n ≤ 2d + 2, the set of centers of the balls in an optimal conﬁguration can always be obtained as the union of the vertex sets of n − d − 1 mutually orthogonal regular simplices of circumradius one, whose dimensions are as equal as possible.
Th: On the intersection of a convex disc and a [FeT77] G. Fejes To polygon, Acta Math. Acad. Sci. Hungar. 29 (1977) 149–153. ´ th: Covering the plane by convex discs, Acta [FeTG72] G. Fejes To Math. Acad. Sci. Hungar. 23 (1972) 263–270. 22 1 Density Problems for Packings and Coverings ´ th, W. Kuperberg: Thin non-lattice covering [FeTK95] G. Fejes To with an aﬃne image of a strictly convex body, Mathematika 42 (1995) 239–250. ´ th, T. Zamfirescu: For most convex discs [FeTZ94] G. Fejes To thinnest covering is not lattice-like, in: Intuitive Geometry (Szeged, 1991) K.
Algebraic topology: Proc. conf. Goettingen 1984 by Larry Smith