From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments

Quantitative form of Ball's Cube slicing in $\mathbb{R}^n$ and equality cases in the min-entropy power inequality

We prove a quantitative form of the celebrated Ball's theorem on cube slicing in $\mathbb{R}^n$ and obtain, as a consequence, equality cases in the min-entropy power inequality. Independently, we also give a quantitative form of Khintchine's inequality in the special case $p=1$.

$\\(\ell_p\\)$ 中立方体切片的韧性

Resilience of cube slicing in $\ell_p$

本研究证明 Ball 的立方体切片定理在 $p > 10^{15}$ 时对 $\ell_p$ 球成立，并推广 Szarek 的 Khinchin 不等式至 $q$ 接近 1 的情况，解决了 Ball--Szarek 超平面在特定参数范围内的韧性难题。

Szarek 不等式与 Ball 不等式的分布稳定性

Distributional stability of the Szarek and Ball inequalities

本文证明 Szarek 最优 Khinchin 不等式与 Ball 立方体切片不等式在接近 Rademacher 分布且权重受 1/√2 分数约束时的分布稳定性，采用傅里叶分析方法。

$p > 2$ 时 $l^n_p(\mathbb{C})$ 单位球的极大截面

Maximal sections of the unit ball of $l^n_p(\mathbb{C})$ for $p > 2$

本文研究复 $l^n_p$ 单位球的极大截面，证明多圆盘切片定理的稳健性仅在大 $p$、小 $n$ 时成立，而在任意 $p>2$ 且大 $n$ 时失效。

On sharp bounds for marginal densities of product measures

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if $f$ is a probability density on $\R^n$ of the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$, say bounded by one, then the density of any marginal $π_E(f)$ is bounded by $2^{k/2}$, where $k$ is the dimension of $E$. The proof relies on an adaptation of Ball's approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of suc…