Statistical Papers

, Volume 57, Issue 4, pp 1059–1075 | Cite as

Extremal measures maximizing functionals based on simplicial volumes

  • Luc PronzatoEmail author
  • Henry P. Wynn
  • Anatoly Zhigljavsky
Regular Article


We consider functionals measuring the dispersion of a d-dimensional distribution which are based on the volumes of simplices of dimension \(k\le d\) formed by \(k+1\) independent copies and raised to some power \(\delta \). We study properties of extremal measures that maximize these functionals. In particular, for positive \(\delta \) we characterize their support and for negative \(\delta \) we establish connection with potential theory and motivate the application to space-filling design for computer experiments. Several illustrative examples are presented.


Potential theory Logarithmic potential Computer experiments Space-filling design 

Mathematics Subject Classification

62K05 31C15 



The work of the first author was partly supported by the ANR project 2011-IS01-001-01 DESIRE (DESIgns for spatial Random fiElds). The third author was supported by the Russian Science Foundation, project Nb. 15-11-30022 “Global optimization, supercomputing computations, and application”.


  1. Audze P, Eglais V (1977) New approach for planning out of experiments. Probl Dyn Strengths 35:104–107Google Scholar
  2. Björck G (1956) Distributions of positive mass, which maximize a certain generalized energy integral. Arkiv för Matematik 3(21):255–269MathSciNetCrossRefzbMATHGoogle Scholar
  3. Fedorov VV (1972) Theory of optimal experiments. Academic Press, New YorkGoogle Scholar
  4. Hardin DP, Saff EB (2004) Discretizing manifolds via minimum energy points. Notices AMS 51(10):1186–1194MathSciNetzbMATHGoogle Scholar
  5. Johnson ME, Moore LM, Ylvisaker D (1990) Minimax and maximin distance designs. J Stat Plan Inference 26:131–148MathSciNetCrossRefGoogle Scholar
  6. Landkof NS (1972) Foundations of modern potential theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245MathSciNetzbMATHGoogle Scholar
  8. Morris MD, Mitchell TJ (1995) Exploratory designs for computational experiments. J Stat Plan Inference 43:381–402CrossRefzbMATHGoogle Scholar
  9. Pronzato L, Müller WG (2012) Design of computer experiments: space filling and beyond. Stat Comput 22:681–701MathSciNetCrossRefzbMATHGoogle Scholar
  10. Pronzato L, Pázman A (2013) Design of experiments in nonlinear models. Asymptotic normality, optimality criteria and small-sample properties. LNS 212, Springer, New YorkGoogle Scholar
  11. Pronzato L, Wynn HP, Zhigljavsky A (2016) Extended generalised variances, with applications. Bernoulli (to appear). arXiv preprint arXiv:1411.6428
  12. Saff EB (2010) Logarithmic potential theory with applications to approximation theory. Surv Approx Theory 5(14):165–200MathSciNetzbMATHGoogle Scholar
  13. Schilling RL, Song R, Vondracek Z (2012) Bernstein functions: theory and applications. de Gruyter, Berlin/BostonCrossRefzbMATHGoogle Scholar
  14. Zhigljavsky AA, Dette H, Pepelyshev A (2010) A new approach to optimal design for linear models with correlated observations. J Am Stat Assoc 105(491):1093–1103MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Luc Pronzato
    • 1
    Email author
  • Henry P. Wynn
    • 2
  • Anatoly Zhigljavsky
    • 3
    • 4
  1. 1.CNRS, Laboratoire I3S, UMR 7172, UNS, CNRS; 2000route des Lucioles, Les Algorithmes, bât. Euclide BSophia AntipolisFrance
  2. 2.London School of EconomicsLondonUK
  3. 3.School of MathematicsCardiff UniversityCardiffUK
  4. 4.Lobachevsky Nizhny Novgorod State UniversityNizhny NovgorodRussia

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