A generalization of Bang’s lemma

Ambrus, Gergely

doi:10.1090/proc/16228

A generalization of Bang’s lemma
HTML articles powered by AMS MathViewer

by Gergely Ambrus;

Proc. Amer. Math. Soc. 151 (2023), 1277-1284

DOI: https://doi.org/10.1090/proc/16228

Published electronically: December 21, 2022

HTML | PDF | Request permission

Abstract:

We prove a common extension of Bang’s and Kadets’ lemmas for contact pairs, in the spirit of the Colourful Carathéodory Theorem. We also formulate a generalized version of the affine plank problem and prove it under special assumptions. In particular, we obtain a generalization of Kadets’ theorem. Finally, we give applications to problems regarding translative and homothetic coverings.

References

Arseniy Akopyan and Roman Karasev, Kadets-type theorems for partitions of a convex body, Discrete Comput. Geom. 48 (2012), no. 3, 766–776. MR 2957644, DOI 10.1007/s00454-012-9437-1
Arseniy Akopyan, Roman Karasev, and Fedor Petrov, Bang’s problem and symplectic invariants, J. Symplectic Geom. 17 (2019), no. 6, 1579–1611. MR 4057722, DOI 10.4310/JSG.2019.v17.n6.a1
Ralph Alexander, A problem about lines and ovals, Amer. Math. Monthly 75 (1968), 482–487. MR 234351, DOI 10.2307/2314701
G. Ambrus, Appendix: Plank problems, Manuscript, 2010.
Keith Ball, The plank problem for symmetric bodies, Invent. Math. 104 (1991), no. 3, 535–543. MR 1106748, DOI 10.1007/BF01245089
Keith Ball, A lower bound for the optimal density of lattice packings, Internat. Math. Res. Notices 10 (1992), 217–221. MR 1191572, DOI 10.1155/S1073792892000242
Keith Ball, Convex geometry and functional analysis, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 161–194. MR 1863692, DOI 10.1016/S1874-5849(01)80006-1
Thøger Bang, On covering by parallel-strips, Mat. Tidsskr. B 1950 (1950), 49–53. MR 38085
Thøger Bang, A solution of the “plank problem.”, Proc. Amer. Math. Soc. 2 (1951), 990–993. MR 46672, DOI 10.1090/S0002-9939-1951-0046672-4
Thøger Bang, Some remarks on the union of convex bodies, Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Institution, Lund, 1954, pp. 5–11. MR 65183
Alexey Balitskiy, A multi-plank generalization of the Bang and Kadets inequalities, Israel J. Math. 245 (2021), no. 1, 209–230. MR 4357461, DOI 10.1007/s11856-021-2210-5
Imre Bárány, A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), no. 2-3, 141–152. MR 676720, DOI 10.1016/0012-365X(82)90115-7
Peter Brass, William Moser, and János Pach, Research problems in discrete geometry, Springer, New York, 2005. MR 2163782
András Bezdek, On a generalization of Tarski’s plank problem, Discrete Comput. Geom. 38 (2007), no. 2, 189–200. MR 2343303, DOI 10.1007/s00454-007-1333-8
A. Bezdek and K. Bezdek, A solution of Conway’s fried potato problem, Bull. London Math. Soc. 27 (1995), no. 5, 492–496. MR 1338694, DOI 10.1112/blms/27.5.492
A. Bezdek and K. Bezdek, Conway’s fried potato problem revisited, Arch. Math. (Basel) 66 (1996), no. 6, 522–528. MR 1388103, DOI 10.1007/BF01268872
Károly Bezdek, Tarski’s plank problem revisited, Geometry—intuitive, discrete, and convex, Bolyai Soc. Math. Stud., vol. 24, János Bolyai Math. Soc., Budapest, 2013, pp. 45–64. MR 3204554, DOI 10.1007/978-3-642-41498-5_{2}
L. Fejes Tóth, G. Fejes Tóth, and W. Kuperberg, Lagerungen – Arrangements in the plane, on the sphere, and in space, Grundlehren der mathematischen Wissenschaften, Vol. 360, Springer Verlag, 2023.
W. Fenchel, On Th. Bang’s solution of the plank problem, Mat. Tidsskr. B 1951 (1951), 49–51. MR 47347
R. J. Gardner, Relative width measures and the plank problem, Pacific J. Math. 135 (1988), no. 2, 299–312. MR 968614, DOI 10.2140/pjm.1988.135.299
Alexey Glazyrin, Covering a ball by smaller balls, Discrete Comput. Geom. 62 (2019), no. 4, 781–787. MR 4027615, DOI 10.1007/s00454-018-0010-4
Henry F. Hunter, Some special cases of Bang’s inequality, Proc. Amer. Math. Soc. 117 (1993), no. 3, 819–821. MR 1145419, DOI 10.1090/S0002-9939-1993-1145419-5
Vladimir Kadets, Coverings by convex bodies and inscribed balls, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1491–1495. MR 2111950, DOI 10.1090/S0002-9939-04-07650-6
W. O. J. Moser, On the relative widths of coverings by convex bodies, Canad. Math. Bull. 1 (1958), 154, 168, 174. MR 123235, DOI 10.4153/CMB-1958-015-9
Márton Naszódi, Covering a set with homothets of a convex body, Positivity 14 (2010), no. 1, 69–74. MR 2596464, DOI 10.1007/s11117-009-0005-8
Márton Naszódi, Flavors of translative coverings, New trends in intuitive geometry, Bolyai Soc. Math. Stud., vol. 27, János Bolyai Math. Soc., Budapest, 2018, pp. 335–358. MR 3889267
F. L. Nazarov, The Bang solution of the coefficient problem, Algebra i Analiz 9 (1997), no. 2, 272–287 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 407–419. MR 1468554
D. Ohmann, Über die Summe der Inkreisradien bei Überdeckung, Math. Ann. 125 (1953), 350–354 (German). MR 53541, DOI 10.1007/BF01343130
V.P. Soltan, personal communication, 1990.
Valeriu Soltan and Éva Vásárhelyi, Covering a convex body by smaller homothetic copies, Geom. Dedicata 45 (1993), no. 1, 101–113. MR 1199732, DOI 10.1007/BF01667407
A. Tarski, Uwagi o stopniu równowazności wielokatów (Further remarks about the degree of equivalence of polygons). Odbilka Z. Parametru 2 (1932), 310–314.
É. Vásárhelyi, Über eine Überdeckung mit homothetischen Dreiecken, Beitr. Algebra Geom. 17 (1984), 61–70.