Arrangements of Homothets of a Convex Body II
Keywords:convex body, homothets, Minkowski arrangement, packing, covering
AbstractA family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior.
We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most 2*3^d members.
This improves a result of Polyanskii (Discrete Mathematics 340 (2017), 1950--1956).
Using similar ideas, we also give a proof the following result of Polyanskii:
Let K_1,....,K_n be a sequence of homothets of the o-symmetric convex body K, such that for any i<j, the center of K_j lies on the boundary of K_i. Then n<= O(3^d d).
2. K. Böröczky and L. Szabó, Minkowski arrangements of spheres, Monatsh. Math. 141 (2004), no. 1, 11-19. MR 2109518
3. L. Fejes Tóth, Minkowskian distribution of discs, Proc. Amer. Math. Soc. 16 (1965), 999-1004. MR 0180921
4. L. Fejes Tóth, Minkowskian circle-aggregates, Math. Ann. 171 (1967), 97-103. MR 0221386
5. L. Fejes Tóth, Minkowski circle packings on the sphere, Discrete Comput. Geom. 22 (1999), no. 2, 161-166. MR 1698538
6. Z. Füredi and P. A. Loeb, On the best constant for the Besicovitch covering theorem, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1063-1073. MR 1249875 (95b:28003)
7. Z. Lángi and M. Naszódi, On the Bezdek-Pach conjecture for centrally symmetric convex bodies, Canad. Math. Bull. 52 (2009), no. 3, 407-415. MR 2547807
8. M. Naszódi, On a conjecture of Károly Bezdek and János Pach, Period. Math. Hungar. 53 (2006), no. 1-2, 227-230. MR 2286473
9. M. Naszódi, L. Martínez-Sandoval, and S. Smorodinsky, Bounding a global red-blue proportion using local conditions, Proceedings of the 33rd European Workshop on Computational Geometry (EuroCG2017), Malmö University, 2017, pp. 213-217.
10. M. Naszódi, J. Pach, and K. J. Swanepoel, Arrangements of homothets of a convex body, Mathematika 63 (2017), 696-710. MR 3706603
11. A. Polyanskii, Pairwise intersecting homothets of a convex body, Discrete Math. 340 (2017), 1950-1956.
12. K. J. Swanepoel, Combinatorial distance geometry in normed spaces, New trends in intuitive geometry (Gergely Ambrus, Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, and János Pach, eds.), Bolyai Soc. Math. Stud., vol. 27, Springer, 2018, pp. 407-458.
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