On the moduli of convexity and smoothness

Figiel, T.

doi:10.4064/sm-56-2-121-155

Cited by 190 publications

(118 citation statements)

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“…It is known [7,8] [7,8]. It remains to show that if I and J are infinite, then W H is not of cotype q < m. But if I and J are infinite, then W H contains all finite dimensional subspaces of ℓ m as subspace, and thus it cannot be of cotype q < m.…”

Section: Theorem 216mentioning

confidence: 99%

Graph norms and Sidorenko’s conjecture

Hatami

2010

Isr. J. Math.

169

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Let H and G be two finite graphs. Define hH (G) to be the number of homomorphisms from H to G. The function hH (•) extends in a natural way to a function from the set of symmetric matrices to R such that for AG, the adjacency matrix of a graph G, we have hH (AG) = hH (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then hH (•) 1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function hH (•) 1/m is a norm.We prove that hH (•) 1/m is a norm if and only if a Hölder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that hH (•) 1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when hH (•) 1/m is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture.We also investigate the hH (•) 1/m norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.

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Section: Theorem 216mentioning

confidence: 99%

Graph norms and Sidorenko’s conjecture

Hatami

2010

Isr. J. Math.

169

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“…This formula was established in [37] (see also [28]). It gives us a special instance of the following general result proved in [72].…”

Section: It Is Easy To See That If Y Is Finitely Representable Inmentioning

confidence: 99%

“…Another approach to finite dimensional uniform convexity was found by Sullivan [103] who defined the modulus of k-convexity. Theorem 25 was proved by Lin [74], but its partial cases with k = 1, 2 had been earlier obtained in [37] and [40]. The reader should be warned that there are different definitions of k-uniform convexity in the literature (see, for instance, [52], p. 73).…”

Section: Obviously τ Cs(x) = Inf Limmentioning

confidence: 99%

Geometrical Background of Metric Fixed Point Theory

Prus

2001

Handbook of Metric Fixed Point Theory

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“…Since p > q the space X has modulus of convexity of power type p (see [7]) and by Theorem 1-1, there exists a separating polynomial of degree at most p on X. By Lemma 3-3, we may actually assume that there is a p-homogeneous separating polynomial.…”

Section: Proof Ofmentioning

confidence: 99%