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41
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. COMPUT
, 1998
"... It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algori ..."
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Cited by 129 (6 self)
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It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
Property (T) and rigidity for actions on Banach spaces
 BHV] [BoS] [Bou] [BuSc] [BuSc’] [BrSo] [C] M. B. Bekka, P. de la Harpe, Alain Valette. “Kazhdan’s
, 2005
"... Abstract. We study property (T) and the fixed point property for actions on L p and other Banach spaces. We show that property (T) holds when L 2 is replaced by L p (and even a subspace/quotient of L p), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for L ..."
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Cited by 52 (6 self)
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Abstract. We study property (T) and the fixed point property for actions on L p and other Banach spaces. We show that property (T) holds when L 2 is replaced by L p (and even a subspace/quotient of L p), and that in fact it is independent of 1 ≤ p < ∞. We show that the fixed point property for L p follows from property (T) when 1 < p < 2 +ε. For simple Lie groups and their lattices, we prove that the fixed point property for L p holds for any 1 < p < ∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.
Kazhdan and Haagerup properties from the median viewpoint
 Adv. Math
"... Abstract. We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (aTmenability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classi ..."
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Cited by 24 (1 self)
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Abstract. We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (aTmenability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on L pspaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of RobertsonSteger. Contents
Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 19 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
Corrigendum to “Enhanced negative type for finite metric trees
 J. Funct. Anal
"... A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of ..."
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Cited by 18 (11 self)
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A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non trivial 1negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree (T,d) must have strict pnegative type for all p in an open interval (1 − ζ,1 + ζ), where ζ> 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T. In particular, if the edges of the tree are not weighted, then it follows that ζ depends only upon the number of vertices in the tree. We also give an example of an infinite metric tree that has strict 1negative type but does not have pnegative type for any p> 1. This shows that the maximal pnegative type of a metric space can be strict. Key words: Finite metric trees, strict negative type, generalized roundness
Inverse formula for the BlaschkeLevy representation
 Houston J. Math
, 1997
"... Abstract. We say that an even continuous function H on the unit sphere Ω in Rn admits the BlaschkeLevy representation with q> 0 if there exists an even function b ∈ L1(Ω) so that Hq (x) = ∫ Ω (x, ξ)qb(ξ) dξ for every x ∈ Ω. This representation has numerous applications in convex geometry, pro ..."
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Cited by 16 (1 self)
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Abstract. We say that an even continuous function H on the unit sphere Ω in Rn admits the BlaschkeLevy representation with q> 0 if there exists an even function b ∈ L1(Ω) so that Hq (x) = ∫ Ω (x, ξ)qb(ξ) dξ for every x ∈ Ω. This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of H) for calculating b out of H. We use this formula to give a sufficient condition for isometric embedding of a space into Lp which contributes to the 1937 P.Levy’s problem and to the study of zonoids. Another application gives a Fourier transform formula for the volume of (n − 1)dimensional central sections of star bodies in Rn. We apply this formula to find the minimal and maximal volume of central sections of the unit balls of the spaces ℓn p with 0 < p < 2. 1.
definite distributions and subspaces of L−p with applications to stable processes
 Canad. Math. Bull
, 1999
"... Abstract. We define embedding of an ndimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lpspaces. The wellknown connection between embeddings into Lp and positive definite functions is extended to the case of ..."
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Cited by 15 (6 self)
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Abstract. We define embedding of an ndimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lpspaces. The wellknown connection between embeddings into Lp and positive definite functions is extended to the case of negative p by showing that a normed space embeds in L−p if and only if ‖x ‖ −p is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg’s problems by proving that the spaces ℓ n q, 2 < q ≤ ∞ embed in L−p if and only if p ∈ [n − 3, n). We show that the technique of embedding in L−p can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every p ∈ [n−3, n), E(maxi=1,...,n Xi  −p) ≥ E(maxi=1,...,n Yi  −p), where X1,..., Xn and Y1,...,Yn are jointly qstable symmetric random variables, 0 < q ≤ 2, so that, for some k ∈ N, 1 ≤ k < n, the vectors (X1,...,Xk) and (Xk+1,...,Xn) have the same distributions as (Y1,...,Yk) and (Yk+1,...,Yn), respectively, but Yi and Yj are independent for every choice of 1 ≤ i ≤ k, k + 1 ≤ j ≤ n. 1.