On the volume of hyperplane sections of a \(d\)-cube. (English) Zbl 1488.52013
Summary: We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric \(d\)-dimensional cube. This confirms a conjecture posed by Imre Bárány and Péter Frankl.
MSC:
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
References:
| [1] | Aliev, I., Siegel’s lemma and sum-distinct sets, Discrete Comput. Geom., 39, 59-66 (2008) · Zbl 1184.11022 · doi:10.1007/s00454-008-9059-9 |
| [2] | Ball, K., Cube slicing in \(\mathbb{R}^n\), Proc. Amer. Math. Soc., 97, 465-472 (1986) · Zbl 0601.52005 |
| [3] | I. Bárány and P. Frankl, Cells in the box and a hyperplane, manuscript (2020) |
| [4] | Borwein, D.; Borwein, J., Some remarkable properties of sinc and related integrals, Ramanujan J., 5, 73-89 (2001) · Zbl 0991.42004 · doi:10.1023/A:1011497229317 |
| [5] | Chakerian, D.; Logothetti, D., Cube slices, pictorial triangles, and probability, Math. Mag., 64, 219-241 (1991) · Zbl 0754.52002 · doi:10.1080/0025570X.1991.11977612 |
| [6] | R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press (Cambridge, 1995) · Zbl 0864.52001 |
| [7] | A. Koldobsky, Fourier Analysis in Convex Geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society (Providence, 2005) · Zbl 1082.52002 |
| [8] | Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math., 71, 232-261 (1988) · Zbl 0657.52002 · doi:10.1016/0001-8708(88)90077-1 |
| [9] | P. S. Laplace, Théorie Analytique des Probabilités (Paris, 1812) · JFM 18.0166.01 |
| [10] | Medhurst, RG; Roberts, JH, Evaluation of the integral \(I_n(b)=\frac{2}{\pi }\int_0^\infty (\frac{\sin x}{x})^n \cos (bx)\, dx\), Math. Comp., 19, 113-117 (1965) · Zbl 0147.14601 |
| [11] | Pólya, G., Berechnung eines Bestimmten Integrals, Math. Ann., 74, 204-212 (1913) · JFM 44.0357.02 · doi:10.1007/BF01456040 |
| [12] | N. J. A. Sloane, Sequences A049330 and A049331, The On-Line Encyclopedia of Integer Sequences, oeis.org |
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