论文标题
在由log-concove随机向量的加权总和定义的多综合规范上
On a multi-integral norm defined by weighted sums of log-concave random vectors
论文作者
论文摘要
令$ c $和$ k $为$ {\ mathbb r}^n $的集中对称凸面。我们表明，如果$ c $是各向同性的，则\ begin {equination*} \ | {\ bf t} \ | _ {c^s，k} = \ int_ {c} \ cdots \ cdots \ cdots \ int_ {c} c} dx_s \ leq c_1l_c（\ log n）^5 \，\ sqrt {n} m（k）\ | {\ | {\ bf t} \ | _2 \ | _2 \ eend {equication {qore {equation*}每一个$ s \ geq 1 $ and＆$ s \ geq 1 $和$ {\ bf t} = \ bf t} =（ r}^s $，其中$ l_c $是$ c $和$ m（k）的各向同性常数：= \ int_ {s^{n-1}} \ |ξ\ |_kdσ（ξ）$。这将V. Milman的问题降低到了从上方估算的各向同性凸体的$ M（k）$的问题。证明是基于一个观察结果，该观察结合了埃尔丹，lehec和klartag在切片问题上的结果：如果$μ$是$ {\ mathbb r}^n $上的各向异性对数凸概率措施$ i_1（μ，k）：= \ int _ {{\ mathbb r}^n} \ | x \ | | _k \，dμ（x）\ leq c_2 \ sqrt {n} {n}（\ log n）^5 \，m（k）。
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 observation that combines results of Eldan, Lehec and Klartag on the slicing problem: If $μ$ is an isotropic log-concave probability measure on ${\mathbb R}^n$ then, for any centrally symmetric convex body $K$ in ${\mathbb R}^n$ we have that $$I_1(μ,K):=\int_{{\mathbb R}^n}\|x\|_K\,dμ(x)\leq c_2\sqrt{n}(\log n)^5\,M(K).$$ We illustrate the use of this inequality with further applications.
