Skip to main content
Log in

Standard simplices and pluralities are not the most noise stable

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript


The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Mathematische Zeitschrift 185 (1984), 339–353.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Borell, The Brunn–Minkowski inequality in Gauss space, Inventiones Mathematicae 30 (1975), 207–216.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Borell, Geometric bounds on the Ornstein–Uhlenbeck velocity process, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 70 (1985), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Borell, The Ehrhard inequality, Comptes Rendus Mathématique. Académie des Sciences. Paris 337 (2003), 663–666.

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Chavel, Riemannian Geometry, second edition, Cambridge Studies in Advanced Mathematics, Vol. 98, Cambridge University Press, Cambridge, 2006.

  6. R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, Journal für die Reine ud Angewandte Mathematik 611 (2007), 75–108.

    MathSciNet  MATH  Google Scholar 

  7. T. H. Colding and W. P. Minicozzi, II, Minimal surfaces and mean curvature flow, in Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics (ALM), Vol. 20, International Press, Somerville, MA, 2011, pp. 73–143.

    MathSciNet  MATH  Google Scholar 

  8. T. H. Colding and W. P. Minicozzi, II, Generic mean curvature flow I: Generic singularities, Annals of Mathematics 175 (2012), 755–833.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. H. Colding, T. Ilmanen, W. P. Minicozzi, II and B. White, The round sphere minimizes entropy among closed self-shrinkers, Journal of Differential Geometry 95 (2013), 53–69.

    MathSciNet  MATH  Google Scholar 

  10. T. H. Colding and W. P. Minicozzi, II, A Course in Minimal Surfaces, Graduate Studies in Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 2011.

  11. J. Corneli, I. Corwin, S. Hurder, V. Sesum, Y. Xu, E. Adams, D. Davis, M. Lee, R. Visocchi and N. Hoffman, Double bubbles in Gauss space and spheres, Houston Journal of Mathematics 34 (2008), 181–204.

    MathSciNet  MATH  Google Scholar 

  12. R. Eldan, A two-sided estimate for the gaussian noise stability deficit, Inventiones Mathematicae 201 (2015), 561–624.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Heilman, Euclidean partitions optimizing noise stability, Electroninc Journal of Probability 19 (2014), 37 pp.

  14. M. Hutchings, F. Morgan, M. Ritoré and A. Ros, Proof of the double bubble conjecture, Annals of Mathematics 155 (2002), 459–489.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Isaksson and E. Mossel, Maximally stable Gaussian partitions with discrete applications, Israel Journal of Mathematics 189 (2012), 347–396.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Kalai, A Fourier-theoretic perspective on the Condorcet paradox and Arrow’s theorem, Advances in Applied Mathematics 29 (2002), 412–426.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Khot, G. Kindler, E. Mossel and R. O’Donnell, Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?, SIAM Journal on Computing 37 (2007), 319–357.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Khot and A. Naor, Approximate kernel clustering, Mathematika 55 (2009), 129–165.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Kindler and R. O’Donnell, Gaussian noise sensitivity and Fourier tails, in 2012 IEEE 27th Conference on Computational Complexity—CCC 2012, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 137–147.

    Chapter  Google Scholar 

  20. M. Ledoux, Isoperimetry and Gaussian analysis, in Lectures on Probability Theory and Statistics (Saint-Flour, 1994), Lecture Notes in Mathematics, Vol. 1648, Springer, Berlin, 1996, pp. 165–294.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Lévy, Problèmes concrets d’analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino, Gauthier-Villars, Paris, 1951.

    MATH  Google Scholar 

  22. F. Morgan, Geometric Measure Theory, fourth edition, Elsevier/Academic Press, Amsterdam, 2009.

    MATH  Google Scholar 

  23. E. Mossel and J. Neeman, Robust optimality of gaussian noise stability, Journal of the European Mathematical Society 17 (2015), 433–482.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality, Annals of Mathematics 171 (2010), 295–341.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003.

    Google Scholar 

  26. J. Steiner, Einfache beweise der isoeperimetrische hauptsätze, Journal für die Reine und Angewandte Mathematik 18 (1838), 281–296.

    Article  Google Scholar 

  27. V. N. Sudakov and B. S. Cirel’son, Extremal properties of half-spaces for spherically invariant measures. Problems in the theory of probability distributions, II, Zapiski Naučnyh Seminarov Leningradsogo Otdelenija Matematičeskogo Instituta im V. A. Steklova 41 (1974), 14–24, 165.

    MathSciNet  Google Scholar 

  28. J. M. Sullivan and F. Morgan, Open problems in soap bubble geometry, International Journal of Mathematics 7 (1996), 833–842.

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Weierstrass, Mathematische Werke. VII. Vorlesungen über Variationsrechnung, Georg Olms Verlagsbuchhandlung, Hildesheim, 1967.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Steven Heilman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heilman, S., Mossel, E. & Neeman, J. Standard simplices and pluralities are not the most noise stable. Isr. J. Math. 213, 33–53 (2016).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: