Abstract
AbstractThe (real) Grothendieck constant ${K}_{G} $ is the infimum over those $K\in (0, \infty )$ such that for every $m, n\in \mathbb{N} $ and every $m\times n$ real matrix $({a}_{ij} )$ we have $$\begin{eqnarray*}\displaystyle \max _{\{ x_{i}\} _{i= 1}^{m} , \{ {y}_{j} \} _{j= 1}^{n} \subseteq {S}^{n+ m- 1} }\sum _{i= 1}^{m} \sum _{j= 1}^{n} {a}_{ij} \langle {x}_{i} , {y}_{j} \rangle \leqslant K\max _{\{ \varepsilon _{i}\} _{i= 1}^{m} , \{ {\delta }_{j} \} _{j= 1}^{n} \subseteq \{ - 1, 1\} }\sum _{i= 1}^{m} \sum _{j= 1}^{n} {a}_{ij} {\varepsilon }_{i} {\delta }_{j} . &&\displaystyle\end{eqnarray*}$$ The classical Grothendieck inequality asserts the nonobvious fact that the above inequality does hold true for some $K\in (0, \infty )$ that is independent of $m, n$ and $({a}_{ij} )$. 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 ${K}_{G} $ remains a mystery. The last progress on this problem was in 1977, when Krivine proved that ${K}_{G} \leqslant \pi / 2\log (1+ \sqrt{2} )$ 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 $({a}_{ij} )$ which exhibit (asymptotically, as $m, n\rightarrow \infty $) a lower bound on ${K}_{G} $ that matches Krivine’s bound. Here, we obtain an improved Grothendieck inequality that holds for all matrices $({a}_{ij} )$ and yields a bound ${K}_{G} \lt \pi / 2\log (1+ \sqrt{2} )- {\varepsilon }_{0} $ for some effective constant ${\varepsilon }_{0} \gt 0$. 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 two-dimensional projections, when combined with a careful partition of ${ \mathbb{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 Frieze–Kannan Cut Norm problem, a generic and well-studied optimization problem with many applications.
Highlights
In his 1953 Resume [14], Grothendieck proved a theorem that he called ‘le theoreme fondamental de la theorie metrique des produits tensoriels’
Rather than analyzing a specific example of matrices, as required by an affirmative solution of Krivine’s conjecture, here we find a new method that performs better than Krivine’s method on all matrices
One should work instead with the optimal partition of R2, which we conjecture is the tiger partition. This too would not be very meaningful, since we do not see a good reason why two-dimensional partitions are special in the present context. It was shown in [30] that by working with Krivine-type rounding schemes in Rk one can approach the exact value of the Grothendieck constant as k → ∞
Summary
In his 1953 Resume [14], Grothendieck proved a theorem that he called ‘le theoreme fondamental de la theorie metrique des produits tensoriels’. Following the upper bounds on KG obtained in [14, 26, 36] (see the alternative proofs of (1.1) in [6, 10, 18, 28, 29, 31], yielding worse bounds on KG), progress on Grothendieck’s Problem 3 halted after a beautiful 1977 theorem of Krivine [23], who proved that One reason for this lack of improvement since 1977 is that Krivine conjectured [23] that his bound is the exact value of KG. Krivine’s conjecture corresponds to a natural geometric intuition about the worst spherical configuration for Grothendieck’s inequality This geometric picture has been crystallized and cleanly formulated as an extremal analytic/geometric problem due to the works of Haagerup, Konig, and Tomczak-Jaegermann.
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