NDLI: Standard simplices and pluralities are not the most noise stable

Content Provider	Springer Nature Link
Author	Heilman, Steven Mossel, Elchanan Neeman, Joe
Copyright Year	2016
Abstract	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.
Starting Page	33
Ending Page	53
Page Count	21
File Format	PDF
ISSN	00212172
Journal	Israel Journal of Mathematics
Volume Number	213
Issue Number	1
e-ISSN	15658511
Language	English
Publisher	The Hebrew University Magnes Press
Publisher Date	2016-04-15
Publisher Place	Jerusalem
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics Algebra Group Theory and Generalizations Analysis Applications of Mathematics Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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