Extremal arrangements of points in the sphere for weighted cone-volume functionals. (English) Zbl 07745157
Summary: Weighted cone-volume functionals are introduced for the convex polytopes in \(\mathbb{R}^n\). For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived, including extremal properties of the regular polytopes involving the \(L_p\) surface area. Some applications to crystallography and quantum theory are also presented.
MSC:
| 52B11 | \(n\)-dimensional polytopes |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 51Mxx | Real and complex geometry |
References:
| [1] | Berman, J. D.; Hanes, K., Volumes of polyhedra inscribed in the unit sphere in \(E^3\), Math. Ann., 188, 78-84 (1970) · Zbl 0187.19604 |
| [2] | Bezdek, K.; Lángi, Z., Volumetric Discrete Geometry (2019), CRC Press · Zbl 1425.52001 |
| [3] | Bilyk, D.; Ferizovi��, D.; Glazyrin, A.; Matzke, R. W.; Park, J.; Vlasiuk, O., Optimal measures for multivariate geometric potentials (2023) |
| [4] | Böröczky, K. J.; Henk, M., Cone volume measure of general centered convex bodies, Adv. Math., 286, 703-721 (2016) · Zbl 1334.52003 |
| [5] | Donahue, J.; Hoehner, S.; Li, B., The maximum surface area polyhedron with five vertices inscribed in the sphere \(\mathbb{S}^2\), Acta Crystallogr., Sect. A, 77, 1, 67-74 (2021) · Zbl 07872977 |
| [6] | Fejes Tóth, L., Egy gömbfelület befedése egybevágó gömbsüvegekkel’, Mat. Fiz. Lapok, 50, 40-46 (1943) · Zbl 0060.34808 |
| [7] | Fejes Tóth, L., Inequalities concerning polygons and polyhedra, Duke Math. J., 15, 3, 817-822 (1948) · Zbl 0032.12001 |
| [8] | Fejes Tóth, L., An inequality concerning polyhedra, Bull. Am. Math. Soc., 54, 2, 139-146 (1948) · Zbl 0031.27901 |
| [9] | Fejes Tóth, L., The isepiphan problem for n-hedra, Am. J. Math., 70, 174-180 (1948) · Zbl 0038.35603 |
| [10] | Fejes Tóth, L., Extremum properties of the regular polyhedra, Can. J. Math., 2, 22-31 (1950) · Zbl 0035.24501 |
| [11] | Fejes Tóth, L., Lagerungen in der Ebene, auf der Kugel und im Raum, Grundlehren der Mathematischen Wissenschaften, vol. 65 (1953), Springer-Verlag · Zbl 0052.18401 |
| [12] | Fejes Tóth, L., Regular Figures, International Series of Monographs on Pure & Applied Mathematics, vol. 48 (1964), Permagon Press · Zbl 0134.15705 |
| [13] | Freeman, N.; Hoehner, S.; Ledford, J.; Pack, D.; Walters, B., Surface areas of equifacetal polytopes inscribed in the unit sphere \(\mathbb{S}^2\), Involve (2023), in press |
| [14] | Haberl, C.; Lutwak, E.; Yang, D.; Zhang, G., The even Orlicz Minkowski problem, Adv. Math., 224, 2485-2510 (2010) · Zbl 1198.52003 |
| [15] | Haberl, C.; Parapatits, L., Valuations and surface area measures, J. Reine Angew. Math. (Crelles J.), 2014, 687, 225-245 (2014) · Zbl 1295.52018 |
| [16] | He, B.; Leng, G.; Li, K., Projection problems for symmetric polytopes, Adv. Math., 207, 73-90 (2006) · Zbl 1111.52012 |
| [17] | Henk, M.; Linke, E., Cone-volume measures of polytopes, Adv. Math., 253, 50-62 (2014) · Zbl 1308.52007 |
| [18] | Heppes, A., An extremal property of certain tetrahedra, Mat. Lapok, 12, 59-61 (1961), (in Hungarian) · Zbl 0103.15505 |
| [19] | Hoehner, S.; Li, B.; Roysdon, R.; Thäle, C., Asymptotic expected T-functionals of random polytopes with applications to \(L_p\) surface areas (2022), submitted for publication |
| [20] | Horváth, Á. G., Volume of convex hull of two bodies and related problems, (Marston, D. E.; Conder, Antoine Deza; Weiss, Asia Ivić, Discrete Geometry and Symmetry (2018), Springer International Publishing: Springer International Publishing Cham), 201-224 · Zbl 1421.52015 |
| [21] | Horváth, Á. G.; Lángi, Z., Maximum volume polytopes inscribed in the unit sphere, Monatshefte Math., 181, 341-354 (2014) · Zbl 1354.52016 |
| [22] | Jian, H.; Lu, J., Existence of solutions to the Orlicz-Minkowski problem, Adv. Math., 344, 262-288 (2019) · Zbl 1408.35061 |
| [23] | Kabluchko, Z.; Litvak, A. E.; Zaporozhets, D., Mean width of regular polytopes and expected maxima of correlated Gaussian variables, J. Math. Sci., 225, 770-787 (2017) · Zbl 1381.52010 |
| [24] | Kabluchko, Z.; Marynych, A.; Temesvari, D.; Thäle, C., Cones generated by random points on half-spheres and convex hulls of Poisson point processes, Probab. Theory Relat. Fields, 175, 1021-1061 (2019) · Zbl 1428.52007 |
| [25] | Kabluchko, Z.; Temesvari, D.; Thäle, C., Expected intrinsic volumes and facet numbers of random beta-polytopes, Math. Nachr., 292, 1, 79-105 (2019) · Zbl 1414.52003 |
| [26] | Kazakov, M., The Structure of the Real Numerical Range and the Surface Area Quantum Entanglement Measure (December 2018), The University of Guelph, Master’s thesis |
| [27] | Klamkin, M. S.; Tsintsifas, G. A., The circumradius-inradius inequality for a simplex, Math. Mag., 52, 20-22 (1979) · Zbl 0405.51005 |
| [28] | Krammer, G., An extremal property of regular tetrahedra, Mat. Lapok, 12, 54-58 (1961), (in Hungarian) · Zbl 0103.15504 |
| [29] | Linhart, J., Über eine Ungleichung für die Oberfläche und den Umkugelradius eines konvexen Polyeders, Arch. Math., 39, 3, 278-284 (1982) · Zbl 0491.52007 |
| [30] | Litvak, A. E., Around the simplex mean width conjecture, (Analytic Aspects of Convexity. Analytic Aspects of Convexity, Springer INdAM Series, vol. 25 (2018), Springer), 73-84 · Zbl 1402.52009 |
| [31] | Ludwig, M.; Reitzner, M., A classification of \(\text{SL}(n)\) invariant valuations, Ann. Math., 172, 2, 1219-1267 (2010) · Zbl 1223.52007 |
| [32] | Lutwak, E., The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differ. Geom., 38, 131-150 (1993) · Zbl 0788.52007 |
| [33] | Lutwak, E.; Yang, D.; Zhang, G., A new affine invariant for polytopes and Schneider’s projection problem, Trans. Am. Math. Soc., 353, 1767-1779 (2001) · Zbl 0971.52011 |
| [34] | Lutwak, E.; Yang, D.; Zhang, G., Orlicz centroid bodies, J. Differ. Geom., 84, 2, 365-387 (2010) · Zbl 1206.49050 |
| [35] | Lutwak, E.; Yang, D.; Zhang, G., Orlicz projection bodies, Adv. Math., 223, 1, 220-242 (2010) · Zbl 1437.52006 |
| [36] | Makovicky, E.; Balić-Z̆unić, T., New measure of distortion for coordination polyhedra, Acta Crystallogr., Sect. B, 54, 766-773 (1998) |
| [37] | Melnyk, T. W.; Knop, O.; Smith, W. R., Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited, Can. J. Chem., 55, 10, 1745-1761 (1977) · Zbl 0367.52012 |
| [38] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications (2013), Cambridge University Press |
| [39] | Sun, Q.; Xiong, G., Sharp affine isoperimetric inequalities for the Minkowski first mixed volume, Bull. Lond. Math. Soc., 52, 1, 161-174 (2020) · Zbl 1443.52007 |
| [40] | Tanner, R. M., Some content maximizing properties of the regular simplex, Pac. J. Math., 52, 2, 611-616 (1974) · Zbl 0287.52008 |
| [41] | Wieacker, J. A., Einige Probleme der polyedrischen Approximation (1978), Albert-Ludwigs-Universität: Albert-Ludwigs-Universität Freiburg im Breisgau, PhD thesis |
| [42] | Xiong, G., Extremum problems for the cone volume functional of convex polytopes, Adv. Math., 225, 6, 3214-3228 (2010) · Zbl 1213.52011 |
| [43] | Yanping, Z.; Binwu, H., On LYZ’s conjecture for the U-functional, Adv. Appl. Math., 87, 43-57 (2017) · Zbl 1371.52004 |
| [44] | Zou, D.; Xiong, G., The minimal Orlicz surface area, Adv. Appl. Math., 61, 25-45 (2014) · Zbl 1376.52017 |
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.
