Abstract
Let K be a convex body in \({\mathbb {R}}^{3}\). We denote the volume of K by Vol(K) and the diameter of K by Diam(K). In this paper we prove that there exists a linear bijection \(T:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3}\) such that Vol\((TK)\ge \frac{\sqrt{2}}{12}\text {Diam}(TK)^3\) with equality if K is a simplex, which was conjectured by Makai Jr. (Studia Sci Math Hungar 13:19–27, 1978) (see also Behrend (Math Ann 113:713–747, 1937. https://doi.org/10.1007/BF01571662). As a corollary, we prove that any set of non-separable translates in a lattice in \({\mathbb {R}}^{3}\) has density of at least \(\frac{1}{12}\), which is a dual analog of Minkowski’s fundamental theorem. Also we prove that Vol\((K)\ge \frac{1}{12}\omega (K)^3\), where \(K\subset {\mathbb {R}}^{3}\) is a convex body and \(\omega (K)\) is the lattice width of K. Moreover, this estimate is tight for some simplex.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Aliev, A.: New estimates for \(d_{2,1}\) and \(d_{3,2}\). arxiv:2207.09552 (2022)
Alvarez Paiva, J., Balacheff, F., Tzanev, K.: Isosystolic inequalities for optical hypersurfaces. Adv. Math. 301, 934–972 (2016). https://doi.org/10.1016/j.aim.2016.07.003
Ball, K.: An elementary introduction to modern convex geometry. http://library.msri.org/books/Book31/files/ball.pdf (1997)
Barthe, F.: An extremal property of the mean width of the simplex. Math. Ann. 4, 685–693 (1998)
Behrend, F.: Über einige affininvarianten konvexer bereiche. Math. Ann. 113, 713–74 (1937). https://doi.org/10.1007/BF01571662
Ivanov, G.: Tight frames and related geometric problems. Can. Math. Bull. 64(4), 942–96 (2021). https://doi.org/10.4153/S000843952000096X
Kalinin, N.: The newton polygon of a planar singular curve and its subdivision. J. Comb. Theory Ser. A 137, 226–256 (2013)
Kalinin, N.: Tropical approach to nagata’s conjecture in positive characteristic. Discrete Comput. Geom. 58, 158–179 (2013)
Kannan, R., Lovasz, L.: Covering minima and lattice-point-free convex bodies. Ann. Math. 128, 577–602 (1988)
Mahler, K.: Polar analogues of two theorems by minkowski. Bull. Austr. Math. Soc. 11, 121–129 (1974)
Makai, E., Jr.: On the thinnest non-separable lattice of convex bodies. Studia Sci. Math. Hungar. 13, 19–27 (1978)
Makai, E., Martini, H.: Density estimates for k-impassable lattices of balls and general convex bodies in \({\mathbb{R}}^{n}\). arxiv:1612.01307 (2016)
Merino, B., Schymura, M.: On the reverse isodiametric problem and Dvoretzky–Rogers-type volume bounds. RACSAM 114, 136 (2020). https://doi.org/10.1007/s13398-020-00867-7
Schymura, M., González Merino, B.: On densities of lattice arrangements intersecting every i-dimensional affine subspace. Discrete Comput. Geom. 58, 663 (2017). https://doi.org/10.1007/s00454-017-9911-x
Tóth, L.F., Makai, E.: On the thinnest non-separable lattice of convex plates. Studia Sci. Math. Hungar. 9, 191 (1974)
Zhang, G.: Restricted chord projection and affine inequalities. Geom. Dedicata 39(2), 213–22 (1991). https://doi.org/10.1007/BF00182294
Acknowledgements
We would like to thank Nikita Kalinin for the help with the preparation of this article.
Funding
No funding was received to assist with the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Aliev, A. The exact bound for the reverse isodiametric problem in 3-space. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 111 (2024). https://doi.org/10.1007/s13398-024-01607-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-024-01607-x