Version 2 changes:
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(All lemma/theorem/section/etc. numbers are according to the version 1
numbering)
-- Fixed various typos + other minor changes (2/18/2010)
-- Removed Lemma G.5 from the Appendix (it was wrong). The error is
that when you "choose the plane that X and Y lie in before choosing
anything else", what you're left with after conditioning is no
longer Gaussian (see "On the maximal perimeter of a convex set in
R^n with respect to a Gaussian measure" by Fedja Nazarov and "The
Reverse Isoperimetric Problem for Gaussian Measure" by Keith Ball
for more on anticoncentration around the boundary of a convex set
under Gaussian measure). The net effect is that Theorem G.6 in the
Appendix reduces the m^6 dependence of Theorem 8.1 to m^4, not
m^2. (2/18/2010)
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-- An early version of this paper advertised a "new" spectral moment
bound for quadratic forms. This bound turns out to not be new, and
is known as the "Hanson-Wright inequality" (thanks to Assaf Naor
for a relevant reference). The current version correctly
attributes the bound, though we've left our proof in there since it
seems to be different from the other proofs we've found of the
Hanson-Wright inequality. (10/13/2010)