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59
Finite dimensional subspaces of L_p
"... this article, we chose to devote this section to describing the change of densities that arise later. It turns out that the framework in which this technique is most naturally used is that of an L p () space when is a probability. For us there is no loss of generality in restricting to that case si ..."
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this article, we chose to devote this section to describing the change of densities that arise later. It turns out that the framework in which this technique is most naturally used is that of an L p () space when is a probability. For us there is no loss of generality in restricting to that case since the space #
An arithmetic proof of John’s ellipsoid theorem
 Arch. Math. (Basel
"... Abstract. Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part ’ of a generalization of John’s theore ..."
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Abstract. Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part ’ of a generalization of John’s theorem and shows the difficulties of the corresponding ‘easy part’.
Grothendiecktype inequalities in combinatorial optimization
 COMM. PURE APPL. MATH
, 2011
"... We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. ..."
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We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
A Banach space determined by the Weil height
 Acta Arith
"... Abstract. The absolute logarithmic Weil height is well defined on the quotient group Q × /Tor ` Q ×´ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrical ..."
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Abstract. The absolute logarithmic Weil height is well defined on the quotient group Q × /Tor ` Q ×´ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrically isomorphic to a codimension one subspace of L 1 (Y, B, λ), where Y is a certain totally disconnected, locally compact space, B is the σalgebra of Borel subsets of Y, and λ is a certain measure satisfying an invariance property with respect to the absolute Galois group Aut(Q/Q). 1.
Biprojectivity and biflatness for convolution algebras of nuclear operators
 Canad. Math. Bull
"... Abstract. For a locally compact group G, the convolution product on the space N (L p (G)) of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra N (L p (G)) and relate them with some properties of the group G, such as compactness, finiteness, dis ..."
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Abstract. For a locally compact group G, the convolution product on the space N (L p (G)) of nuclear operators was defined by Neufang [11]. We study homological properties of the convolution algebra N (L p (G)) and relate them with some properties of the group G, such as compactness, finiteness, discreteness, and amenability. 1.
SMALL SUBSPACES OF Lp
"... Abstract. We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into ℓp ⊕ ℓ2 or X contains a subspace Y, which is isomorphic to ℓp(ℓ2). We also give an intrinsic characterization of when X embeds into ℓp ⊕ℓ2 in terms of weakly null trees in X or, equivalently, ..."
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Abstract. We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into ℓp ⊕ ℓ2 or X contains a subspace Y, which is isomorphic to ℓp(ℓ2). We also give an intrinsic characterization of when X embeds into ℓp ⊕ℓ2 in terms of weakly null trees in X or, equivalently, in terms of the “infinite asymptotic game ” played in X. This solves problems concerning small subspaces of Lp originating in the 1970’s. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000’s. 1.
The UGC hardness threshold of the Lp Grothendieck problem
"... For p ≥ 2 we consider the problem of, given an n × n matrix A = (ai j) whose diagonal entries vanish, approximating in polynomial time the number n� ..."
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For p ≥ 2 we consider the problem of, given an n × n matrix A = (ai j) whose diagonal entries vanish, approximating in polynomial time the number n�
COARSE EMBEDDABILITY INTO BANACH SPACES
, 2008
"... The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse nonembeddability into ℓ2 implie ..."
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The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse nonembeddability into ℓ2 implies presence of expanderlike structures? (b) To what extent ℓ2 is the most difficult space to embed into?