Author:
Grigory Ivanov
grimivanov@gmail.com
Grigory Ivanov
Pontifícia Universidade Católica do Rio de Janeiro, Departamento de Matematica, Rua Marquês de São Vicente, 225, Edifício Cardeal Leme, sala 862, 22451-900 Gávea, Rio de Janeiro , Brazil
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We study the “no-dimensional” analogue of Helly’s theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly’s theorem, fractional Helly’s theorem, colorful Helly’s theorem, and colorful fractional Helly’s theorem.
The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in [2]. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.
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[1]
K. Adiprasito, I. Bárány, and N. H. Mustafa. Theorems of Carathéodory, Helly, and Tverberg without dimension. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2350–2360. SIAM, Philadelphia, PA, 2019.
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[2]
K. Adiprasito, I. Bárány, N. H. Mustafa, and T. Terpai. Theorems of Carathéodory, Helly, and Tverberg without dimension. Discrete Comput. Geom., 64(2):233–258, 2020.
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[3]
J. Alonso, H. Martini, and S. Wu. On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math., 83(1-2):153–189, 2012.
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I. Bárány, F. Fodor, L. Montejano, D. Oliveros, and A. Pór. Colourful and fractional (𝑝, 𝑞)-theorems. Discrete Comput. Geom., 51(3):628–642, 2014.
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I. Bárány and G. Kalai. Helly-type problems. Bull. Amer. Math. Soc. (N.S.), 59(4):471–502, 2022.
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[10]
G. Ivanov. Approximate Carathéodory’s theorem in uniformly smooth Banach spaces. Discrete Comput. Geom., 66(1):273–280, 2021.
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[11]
G. Ivanov. No-dimension Tverberg’s theorem and its corollaries in Banach spaces of type 𝑝. Bull. Lond. Math. Soc., 53(2):631–641, 2021.
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[12]
G. Ivanov and H. Martini. New moduli for Banach spaces. Ann. Funct. Anal., 8(3):350–365, 2017.
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[13]
G. M. Ivanov. Modulus of supporting convexity and supporting smoothness. Eurasian Math. J., 6(1):26–40, 2015.
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M. Katchalski and A. Liu. A problem of geometry in 𝐑𝑛. Proc. Amer. Math. Soc., 75(2):284–288, 1979.
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R. Schneider. Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia Math. Appl. Cambridge University Press, Cambridge, expanded edition, 2014.
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