Shallow sections of the hypercube. (English) Zbl 1519.52007
It was shown in [J. Moody et al., Fields Inst. Commun. 68, 211–228 (2013; Zbl 1270.52008)] that, given a \(d\)-dimensional ball \(B\) (with \(d\geq 3\)) that shares its center with the hypercube \([0, 1]^d\) and contains the midpoints of the edges of \([0, 1]^d\) in its interior, the largest possible \((d-1)\)-dimensional volume of \(H\cap [0, 1]^d\), where \(H\) is a hyperplane of \({\mathbb R}^d\) tangent to \(B\), is obtained if and only if \(H\) is orthogonal to a diagonal of the hypercube.
In the paper under review, the result from the above paper and, to some extent, the result from H. König [Adv. Math. 376, Article ID 107458, 36 p. (2021; Zbl 1459.52009)], are extended for dimension \(d\geq 5\) by showing that, even if \(B\) – with the same center as that of \([0, 1]^d\) – is only required to contain the centers of the square faces of \([0, 1]^d\) in its interior, the volume of \(H \cap [0, 1]^d\) is still maximal if and only if \(H\) is orthogonal to a diagonal of the hypercube.
In the paper under review, the result from the above paper and, to some extent, the result from H. König [Adv. Math. 376, Article ID 107458, 36 p. (2021; Zbl 1459.52009)], are extended for dimension \(d\geq 5\) by showing that, even if \(B\) – with the same center as that of \([0, 1]^d\) – is only required to contain the centers of the square faces of \([0, 1]^d\) in its interior, the volume of \(H \cap [0, 1]^d\) is still maximal if and only if \(H\) is orthogonal to a diagonal of the hypercube.
Reviewer: Victor V. Pambuccian (Glendale)
MSC:
| 52B11 | \(n\)-dimensional polytopes |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
References:
| [1] | Ball, K., Cube slicing in R^n, Proceedings of the American Mathematical Society, 97, 465-473 (1986) · Zbl 0601.52005 |
| [2] | Barrow, D. L.; Smith, P. W., Spline notation applied to a volume problem, American Mathematical Monthly, 86, 50-51 (1979) · Zbl 0411.51013 · doi:10.1080/00029890.1979.11994730 |
| [3] | Berger, M., Geometry Revealed (2010), Heidelberg: Springer, Heidelberg · Zbl 1232.51001 · doi:10.1007/978-3-540-70997-8 |
| [4] | Deza, A.; Hiriart-Urruty, J-B; Pournin, L., Polytopal balls arising in optimization, Contributions to Discrete Mathematics, 16, 125-138 (2021) · Zbl 1494.52011 |
| [5] | Frank, R.; Riede, H., Hyperplane sections of the n-dimensional cube, American Mathematical Monthly, 119, 868-872 (2012) · Zbl 1264.51015 · doi:10.4169/amer.math.monthly.119.10.868 |
| [6] | Gabriélov, A. M.; Gel’fand, I. M.; Losik, M. V., Combinatorial computation of characteristic classes, Functional Analysis and its Applications, 9, 103-115 (1975) · Zbl 0312.57016 · doi:10.1007/BF01075446 |
| [7] | Gel’fand, I. M.; Goresky, M.; MacPherson, R. D.; Serganova, V., Combinatorial geometries, convex polyhedra and schubert cells, Advances in Mathematics, 63, 301-316 (1987) · Zbl 0622.57014 · doi:10.1016/0001-8708(87)90059-4 |
| [8] | Hensley, D., Slicing the cube in R^nand probability, Proceedings of the American Mathematical Society, 73, 95-100 (1979) · Zbl 0394.52006 |
| [9] | Koldobsky, A., Fourier Analysis in Convex Geometry (2005), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1082.52002 · doi:10.1090/surv/116 |
| [10] | H. König, Non-central sections of the simplex, the cross-polytope and the cube, Advances in Mathematics 376 (2021), Article no. 107458. · Zbl 1459.52009 |
| [11] | König, H.; Koldobsky, A., Volumes of low-dimensional slabs and sections in the cube, Advances in Applied Mathematics, 47, 894-907 (2011) · Zbl 1230.52022 · doi:10.1016/j.aam.2011.05.001 |
| [12] | Laplace, P-S, Œuvres complètes (1886), Paris: Gauthier-Villars, Paris · JFM 18.0166.01 |
| [13] | R. Liu and T. Tkocz, A note on the extremal noncentral sections of the cross-polytope, Advances in Applied Mathematics 118 (2020), Article no. 102031. · Zbl 1440.52015 |
| [14] | Meyer, M.; Pajor, A., Sections of the unit ball of l_p^n, Journal of Functional Analysis, 80, 109-123 (1988) · Zbl 0667.46004 · doi:10.1016/0022-1236(88)90068-7 |
| [15] | J. Moody, C. Stone, D. Zach and A. Zvavitch, A remark on extremal non-central sections of the unit cube, in Asymptotic Geometric Analysis, Fields Institute Communications, Vol. 68, Springer, New York, 2013, pp. 211-228. · Zbl 1270.52008 |
| [16] | Pólya, G., Berechnung eines bestimmten Integrals, Mathematische Annalen, 74, 204-212 (1913) · JFM 44.0357.02 · doi:10.1007/BF01456040 |
| [17] | Stanley, R., Eulerian partitions of a unit hypercube, Higher Combinatorics, 49 (1977), Dordrecht: Springer, Dordrecht · Zbl 0359.05001 |
| [18] | Webb, S., Central slices of the regular simplex, Geometriae Dedicata, 61, 19-28 (1996) · Zbl 0853.52007 · doi:10.1007/BF00149416 |
| [19] | Zong, C., The Cube, a Window to Convex and Discrete Geometry (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1100.52002 · doi:10.1017/CBO9780511543173 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
