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In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}/sup n/ /spl rarr/ {-1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical… (More)

Let a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let µ(dx) = c n r exp(−(|x1| r +|x2| r +.. .+|xn| r))dx1dx2. .. dxn be a probability measure on the Euclidean space (R n , ·). We prove that there exists a universal constant C such that for any smooth real function f on R n and any p ∈ [1, 2) Eµf 2 − (Eµ|f | p) 2/p ≤ C(2 − p) a Eµ∇f 2. We prove… (More)

A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ R n with inradius w and γn(K) ≤ 1/2 there is γn(sK) ≤ (2s) w 2 /4 γn(K) for any s ∈ [0, 1]. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm |X| of an isotropic log-concave random vector X ∈ R n , stating that for every t ≥ 1, P ` |X| ≥ ct √ n ´ ≤ exp(−t √ n). More precisely we show that for any log-concave random vector X and any p ≥ 1, (E|X| p) 1/p ∼ E|X| + sup z∈S n−1 (E||z, X| p) 1/p .

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)

- Marek Pycia, Eddie Dekel, Jeffrey Ely, Amanda Friedenberg, Hugo Hopenhayn, Ali Hortacsu +7 others
- 2015

Pay-as-bid is the most popular auction format for selling treasury securities. We prove the uniqueness of pure-strategy Bayesian-Nash equilibria in pay-as-bid auctions where symmetrically-informed bidders face uncertain supply, and we establish a tight sufficient condition for the existence of this equilibrium. Equilibrium bids have a convenient separable… (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)

- Radosław Adamczak, Olivier Guédon, Rafał Latała, Alexander E. Litvak, Krzysztof Oleszkiewicz, Alain Pajor +1 other
- 2012

J o u r n a l o f P r o b a b i l i t y Electron. Abstract Let p ≥ 1, ε > 0, r ≥ (1 + ε)p, and X be a (−1/r)-concave random vector in R n with Euclidean norm |X|. We prove that (E|X| p) 1/p ≤ c (C(ε)E|X| + σp(X)) , where σp(X) = sup |z|≤1 (E||z, X| p) 1/p , C(ε) depends only on ε and c is a universal constant. Moreover, if in addition X is centered then… (More)

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