We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by t...
The Koml\'os problem asks for signs $\varepsilon\in\{-1,1\}^n$ with $\|A\varepsilon\|_\infty$ bounded by an absolute constant whenever the columns of $A\in\mathbb R^{m\times n}$ have Euclidean norm at most $1$. We give the first polynomial-time algorithm for this problem in the unit-cost real-RAM model. Our determinist...
Let $G$ be the Boolean hypercube which carries uniform measure $\lambda$, and let $T_\mu$ denote convolution by a finite positive measure $\mu$ on $G$. For $\psi_\mu(u)=\sup\{u\lambda(\{T_\mu f\geq u\}):f\geq 0,\|f\|_1=1\},$ we prove Talagrand's convolution conjecture (Talagrand, 1989): if $\mu_a=((1+a)\delta_1/2+(1-a)...
Every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size.
Kalai's cube--simplex conjecture asserts that for all positive integers $\ell,k$, there is an integer $f(\ell,k)$ such that every polytope of dimension at least $f(\ell,k)$ has either a simplex $\ell$-face or a cube $k$-face; let $f_s(\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\ell,k...
J. De Loera, Ethan X. Fang, Sheng Guo et al.· 0 citations
We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric wh...
Sheng Guo, Ethan X. Fang, Jun-Wei Lu· 0 citations
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