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In this paper we study functions with low influences on product probability spaces. These are functions f : Ω 1 × · · · × Ω n → R that have E[Var Ωi [f ]] small compared to Var[f ] for each i. The analysis of boolean functions f : {−1, 1} n → {−1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by… (More)

Let S = a 1 r 1 + a 2 r 2 + · · · + a n r n be a weighted Rademacher sum. Friedgut, Kalai, and Naor have shown that if Var(|S|) is much smaller than Var(S), then the sum is largely determined by one of the summands. We provide a simple and elementary proof of this result, strengthen it, and extend it in various ways to a more general setting.

We consider polytopes in R n that are generated by N vectors in R n whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random ±1 poly-topes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is… (More)

We study contraction under a Markov semi-group and influence bounds for functions in L 2 tail spaces, i.e. functions all of whose low level Fourier coefficients vanish. It is natural to expect that certain analytic inequalities are stronger for such functions than for general functions in L 2. In the positive direction we prove an L p Poincaré inequality… (More)

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