×

Mean width of regular polytopes and expected maxima of correlated Gaussian variables. (English. Russian original) Zbl 1381.52010

J. Math. Sci., New York 225, No. 5, 770-787 (2017); translation from Zap. Nauchn. Semin. POMI 442, 75-96 (2015).
The authors study the question of how to arrange \(n+1\) points on the \((n-1)\)-dimensional unit sphere so as to maximize the mean width of their convex hull. An old conjecture states that the arrangement must be regular. The authors use the fact that the mean width is a multiple of the first intrinsic volume, \(V_1\), of the body and reformulate the conjecture as \[ \sup_{x_1, \ldots, x_{n+1} \in \mathbb{S}^{n-1} } V_1( \mathrm{conv}(x_1, \ldots, x_{n+1}) ) = V(T_n), \] in which \(T_n\) is a regular simplex with \(n+1\) vertices inscribed in the sphere \(\mathbb{S}^{n-1}\) and \(\mathrm{conv}\) denotes the convex hull of the set of points. In Theorem 1.2 it is shown that this conjecture holds asymptotically. Moreover, Corollary 2.2 and Corollary 2.4 settle a special case and explain how the first intrinsic volume of the crosspolytope \(C_n\) with \(2n\) vertices is related to the first intrinsic volume of the regular simplex \(T_{2n-1}\) with the same number of vertices. Finally, the authors apply their results to regular polytopes and establish several formulas that were conjectured by Finch.

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
60D05 Geometric probability and stochastic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] R. Adler and J. Taylor, Random Fields and Geometry, Springer-Verlag, New York (2009). · Zbl 1149.60003
[2] F. Affentranger and R. Schneider, “Random projections of regular simplices,” Discrete Comput. Geom., 7, 219-226 (1992). · Zbl 0751.52002 · doi:10.1007/BF02187839
[3] A. Balakrishnan, “Research problem No. 9: Geometry,” Bull. Amer. Math. Soc., 69, 737-738 (1963). · doi:10.1090/S0002-9904-1963-11017-4
[4] A. Balakrishnan, “Signal selection for space communication channels,” in: A. Balakrishnan (ed.), Advances in Communication Systems, Academic Press, New York (1965), pp. 1-31. Bentkus et al. (2007) Bentkus, Jing, Shao, and Zhou. V. Bentkus, B.-Y. Jing, Q.-M. Shao, and W. Zhou, “Limiting distributions of the non-central <Emphasis Type=”Italic“>t-statistic and their applications to the power of <Emphasis Type=”Italic“>t-tests under non-normality,” Bernoulli, 13, 346-364 (2007). · Zbl 1129.60021
[5] A. Bulinski and A. Shashkin, Limit Theorems for Associated Random Fields and Related Systems, World Scientific, Singapore (2007). · Zbl 1154.60037 · doi:10.1142/6555
[6] S. Chatterjee, “An error bound in the Sudakov-Fernique inequality,” arXiv:0510424 (2005).
[7] S. Chatterjee, Superconcentration and Related Topics, Springer International Publishing, Switzerland (2014). · Zbl 1288.60001 · doi:10.1007/978-3-319-03886-5
[8] T. M. Cover and B. Gopinath, Open Problems in Communication and Computation, Springer-Verlag, New York (1987). · Zbl 0628.68001 · doi:10.1007/978-1-4612-4808-8
[9] V. H. de la Peña, T. L. Lai, and Q.-M. Shao, Self-normalized Processes. Limit Theory and Statistical Applications, Springer, Berlin (2009). · Zbl 1165.62071 · doi:10.1007/978-3-540-85636-8
[10] S. R. Finch, “Mean width of a regular cross-polytope,” arXiv:1112.0499 (2011).
[11] S. R. Finch, “Mean width of a regular simplex,” arXiv:1111.4976 (2011). · Zbl 0176.48804
[12] S. R. Finch, “Width distributions for convex regular polyhedra,” arXiv:1110.0671 (2011).
[13] E. Gilbert, “A comparison of signalling alphabets,” Bell Syst. Tech. J., 31, 504-522 (1952). · doi:10.1002/j.1538-7305.1952.tb01393.x
[14] P. Gritzmann and V. Klee, “On the complexity of some basic problems in computational convexity. II: Volume and mixed volumes,” in: T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivíc Weiss (eds.), Polytopes: Abstract, Convex and Computational, Springer Science+Business Media, Dordrecht (1994), pp. 373-466. · Zbl 0819.52008
[15] K. Joag-Dev and F. Proschan, “Negative association of random variables with applications,” Ann. Stat., 11, 286-295 (1983). · Zbl 0508.62041 · doi:10.1214/aos/1176346079
[16] M. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York-Berlin (1983). Omey and Van Gulck(2009). E. Omey and S. Van Gulck, “Domains of attraction of the real random vector (<Emphasis Type=”Italic“>X, X2) and applications,” Publ. Inst. Math., Nouv. Sér., 86, 41-53 (2009). · Zbl 1265.60011
[17] G. Paouris and P. Pivovarov, “Small-ball probabilities for the volume of random convex sets,” Discrete Comput. Geom., 49, 601-646 (2013). · Zbl 1273.52007 · doi:10.1007/s00454-013-9492-2
[18] G. Paouris, P. Pivovarov, and J. Zinn, “A central limit theorem for projections of the cube,” Probab. Theory Related Fields, 159, 701-719 (2014). · Zbl 1301.52017 · doi:10.1007/s00440-013-0518-8
[19] J. Pickands, “Moment convergence of sample extremes,” Ann. Math. Stat., 39, 881-889 (1968). · Zbl 0176.48804 · doi:10.1214/aoms/1177698320
[20] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge (1993). · Zbl 0798.52001 · doi:10.1017/CBO9780511526282
[21] V. Sudakov, “Geometric problems in the theory of infinite-dimensional probability distributions,” Trudy Mat. Inst. AN SSSR, 141, 3-191 (1976). · Zbl 0409.60005
[22] B. Tsirelson, “A geometric approach to maximum likelihood estimation for an infinitedimensional Gaussian location. II,” Teor. Veroyatn. Primen., 30, 772-779 (1985). · Zbl 0604.62081
[23] C. Weber, Elements of Detection and Signal Design, Springer-Verlag, New York (1987). · Zbl 0624.94002 · doi:10.1007/978-1-4612-4774-6
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.