Hyperplane conjecture for quotient spaces of $L_p$

Comments on the floating body and the hyperplane conjecture

We provide a reformulation of the hyperplane conjecture (the slicing problem) in terms of the floating body and give upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\ and the convex floating body $K_δ$ inside $K$.

The variance conjecture on hyperplane projections of l_p^n balls

We show that for any $1\leq p\leq\infty$, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of $\ell_p^n$ verify the variance conjecture $$ \textrm{Var}\,|X|^2\leq C\max_{ξ\in S^{n-1}}\mathbb{E}\langle X,ξ\rangle^2\mathbb{E}|X|^2, $$ where $C$ depends on $p$ but not on the dimension $n$ or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic convex body and the variance conjecture for a random vector uniformly distributed on any Steiner symmetrization of it. As a…

The variance conjecture on some polytopes

We show that any random vector uniformly distributed on any hyperplane projection of $B_1^n$ or $B_\infty^n$ verifies the variance conjecture $$\text{Var}|X|^2\leq C\sup_{ξ\in S^{n-1}}\E<X,ξ>^2\E|X|^2.$$ Furthermore, a random vector uniformly distributed on a hyperplane projection of $B_\infty^n$ verifies a negative square correlation property and consequently any of its linear images verifies the variance conjecture.

两个超平面的Ramos猜想反例

Counterexamples to the Ramos conjecture for two hyperplanes

本文构造反例否证了关于两个超平面的Ramos猜想，并确定了保证多个测度存在公共二超平面等分的最小维数，同时揭示了与相位恢复中正定测量数目的联系。

Kalton-Peck空间与其超空间不同构

Kalton-Peck space is not isomorphic to its hyperplanes

该研究证明实Kalton-Peck空间$Z_2$的超空间均无复结构且$Z_2$满足Ferenczi-Galego偶性，进而得出$Z_2$与其超空间线性不同构，解决了Banach空间理论的一个长期猜想，论证用到多个相关定理并包含自包含证明。

Polydisc version of Arveson's conjecture

In the present paper, we solve the polydisc-version of Arveson Conjecture by giving a complete criteria for essential normality of homogeneous quotient modules of the Hardy module over the polydisc, and it turns out that our method applies to quotient modules of the weighted Bergman modules $A^2_s(\mathbb D^d)$.