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Edgeisoperimetric inequalities and influences
 In Combinatorics, Probability, and Computing
, 2006
"... Abstract We give a combinatorial proof of the result of Kahn, Kalai, and Linial [19], which statesthat every balanced boolean function on the ndimensional boolean cube has a variable with influence of at least \Omega i log nn j. The methods of the proof are then used to recover additional isoperime ..."
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Abstract We give a combinatorial proof of the result of Kahn, Kalai, and Linial [19], which statesthat every balanced boolean function on the ndimensional boolean cube has a variable with influence of at least \Omega i log nn j. The methods of the proof are then used to recover additional isoperimetric results forthe cube, with improved constants. We also state some conjectures about optimal constants and discuss their possible implications.
Hypergraphs, Entropy and Inequalities
 In Preparation
"... reasonable to assume that most mathematicians would be puzzled to find these three terms as, say, key words for the same mathematical paper. (Just in case this puzzlement is a result of being unfamiliar with the term “hypergraph”: a hypergraph is nothing other than a family of sets, and will be defi ..."
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reasonable to assume that most mathematicians would be puzzled to find these three terms as, say, key words for the same mathematical paper. (Just in case this puzzlement is a result of being unfamiliar with the term “hypergraph”: a hypergraph is nothing other than a family of sets, and will be defined formally later.) To further pique the
Hypercontractive Inequality for PseudoBoolean Functions of Bounded Fourier Width
, 2011
"... A function f: {−1, 1} n → R is called pseudoBoolean. It is wellknown that each pseudoBoolean function f can be written as f(x) = I∈F ˆ f(I)χI(x), where F ⊆ {I: I ⊆ [n]}, [n] = {1, 2,..., n}, and χI(x) = ∏ i∈I xi and ˆ f(I) are nonzero reals. The degree of f is max{I : I ∈ F} and the width of ..."
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A function f: {−1, 1} n → R is called pseudoBoolean. It is wellknown that each pseudoBoolean function f can be written as f(x) = I∈F ˆ f(I)χI(x), where F ⊆ {I: I ⊆ [n]}, [n] = {1, 2,..., n}, and χI(x) = ∏ i∈I xi and ˆ f(I) are nonzero reals. The degree of f is max{I : I ∈ F} and the width of f is the minimum integer ρ such that every i ∈ [n] appears in at most ρ sets in F. For i ∈ [n], let xi be a random variable taking values 1 or −1 uniformly and independently from all other variables xj, j = i. Let x = (x1,..., xn). The pnorm of f is fp = (E[f(x)  p]) 1/p for any p ≥ 1. It is wellknown that fq ≥ fp whenever q> p ≥ 1. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on f: if f is of de) d/2 ≤ fp. This inequality is gree d and q> p> 1 then fq q−1 p−1 called the Hypercontractive Inequality. We show that one can replace d by ρ in the Hypercontractive Inequality for each q> p ≥ 2 as follows: fq ≤ ((2r)!ρ r−1) 1/(2r) fp, where r = ⌈q/2⌉. For the case q = 4 and p = 2, which is important in many applications, we prove a stronger inequality: f4 ≤ (2ρ + 1) 1/4 f2. 1