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The Grothendieck constant is strictly smaller than Krivine’s bound
 IN 52ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE. PREPRINT AVAILABLE AT HTTP://ARXIV.ORG/ABS/1103.6161
, 2011
"... The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical Groth ..."
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The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical Grothendieck inequality asserts the nonobvious fact that the above inequality does hold true for some K ∈ (0, ∞) that is independent of m, n and (aij). Since Grothendieck’s 1953 discovery of this powerful theorem, it has found numerous applications in a variety of areas, but despite attracting a lot of attention, the exact value of the Grothendieck constant KG remains a mystery. The last progress on this problem was in π 1977, when Krivine proved that KG � and conjectured that his bound is optimal. Krivine’s conjecture has been restated repeatedly since 1977, resulting in focusing the subsequent research on the search for examples of matrices (aij) which exhibit (asymptotically, as m, n → ∞) a lower bound on KG that matches Krivine’s bound. Here we obtain an improved Grothendieck inequality that holds for all matrices (aij) and yields a bound KG < π 2 log(1+ √ 2) − ε0 for some effective constant ε0> 0. Other than disproving Krivine’s conjecture, and along the way also disproving an intermediate conjecture of König that was made in 2000 as a step towards Krivine’s conjecture, our main contribution is conceptual: despite dealing with a binary rounding problem, random 2dimensional projections, when combined with a careful partition of R 2 in order to round the projected vectors to values in {−1, 1}, perform better than the ubiquitous random hyperplane technique. By establishing the usefulness of higher dimensional rounding schemes, this fact has consequences in approximation algorithms. Specifically, it yields the best known polynomial time approximation algorithm for the FriezeKannan Cut Norm problem, a generic and wellstudied optimization problem with many applications.
Efficient Rounding for the Noncommutative Grothendieck Inequality [Extended Abstract] ∗
"... The classical Grothendieck inequality has applications to the design of approximation algorithms for NPhard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main res ..."
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The classical Grothendieck inequality has applications to the design of approximation algorithms for NPhard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a constantfactor polynomial time approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principle component analysis and the orthogonal Procrustes problem. Categories and Subject Descriptors G.1.6 [Numerical Analysis]: Optimization—Quadratic programming methods
SOLUTION OF THE PROPELLER CONJECTURE IN R³
"... It is shown that every measurable partition {A1,..., Ak} of R 3 satisfies k∑ ..."
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It is shown that every measurable partition {A1,..., Ak} of R 3 satisfies k∑
SOLUTION OF THE PROPELLER CONJECTURE IN R 3
"... Abstract. It is shown that every measurable partition {A1,..., Ak} of R 3 satisfies k∑ ..."
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Abstract. It is shown that every measurable partition {A1,..., Ak} of R 3 satisfies k∑