Concave functions of positive operators, sums and congruences

Tail estimates for norms of sums of log-concave random vectors

We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity $A_{k,m}$ that controls uniformly the operator norm of the sub-matrices with $k$ rows and $m$ columns of a matrix $A$ with independent isotropic log-con…

Norms of weighted sums of log-concave random vectors

Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to "randomized" vector balancing problems.

On a multi-integral norm defined by weighted sums of log-concave random vectors

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb R}^n$. We show that if $C$ is isotropic then \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s \leq c_1L_C(\log n)^5\,\sqrt{n}M(K)\|{\bf t}\|_2\end{equation*} for every $s\geq 1$ and ${\bf t}=(t_1,\ldots ,t_s)\in {\mathbb R}^s$, where $L_C$ is the isotropic constant of $C$ and $M(K):=\int_{S^{n-1}}\|ξ\|_Kdσ(ξ)$. This reduces a question of V.~Milman to the problem of estimating from above the parameter $M(K)$ of an isotropic convex body. The proof is based on an observati…

More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possibl…

α-凹函数的α-卷积的积分不等式

Integral inequalities for $α$-convolutions of $α$-concave functions

该研究针对α-凹函数的α-卷积，建立了Plünnecke–Ruzsa型不等式等积分不等式，衔接了对数凹与拟凹相关研究，丰富了凸分析领域的积分不等式理论。