A note on Bourgain-Milman's universal constant

Bergman kernels for Paley-Wiener spaces and Nazarov's proof of the Bourgain-Milman theorem

We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for Hörmander's estimates for the $\dbar$-equation.

Complex integrals and Kuperberg's proof of the Bourgain-Milman theorem

We give a proof of the Bourgain-Milman theorem using complex methods. The proof is inspired by Kuperberg's, but considerably shorter.

关于Bourgain切片问题的一个注记

A note on Bourgain's slicing problem

本文研究Bourgain切片问题，沿近十年研究路径证明了n≥3时切片常数$L_n$可被通用常数C与$\log(\log n)$的乘积界定。

The random paving property for uniformly bounded matrices

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systemati…

An example related to the slicing inequality for general measures

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{ξ\in S^{n-1}} \int_{K\cap ξ^\bot} f $$ for all centrally-symmetric convex bodies $K$ in $\mathbb{R}^n$ and all even, continuous probability densities $f$ on $K$. Here $|K|$ is the volume of $K$. It was proved by the second-named author that $S_n\le 2\sqrt{n}$, and in analogy with Bourgain's slicing problem, it was asked whether $S_n$ is bounded from above by a universal constant. In this note we construct an example showing that $S_n\ge c\sqrt{n}/\sqrt{\log …