Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNorms of weighted sums of log-concave random vectors
130
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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}cdotsint_{C}Big|sum_{j=1}^st_jx_jBig|_K,dx_1cdots dx_send{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.
rate research
Read More
Our starting point is an improved version of a result of D. Hajela related to a question of Koml{o}s: we show that if $f(n)$ is a function such that $limlimits_{ntoinfty }f(n)=infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $ngeqslant n_0$ and any $Ssubseteq {-1,1}^n$ with cardinality $|S|leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,ldots ,x_nin {mathbb R}^n$ that satisfy $$|epsilon_1x_1+cdots +epsilon_nx_n|_{infty }geqslant csqrt{log f(n)}$$ for all $(epsilon_1,ldots ,epsilon_n)in S$. We obtain analogous results in the case where $x_1,ldots ,x_n$ are independent random points uniformly distributed in the Euclidean unit ball $B_2^n$ or any symmetric convex body, and the $ell_{infty }^n$-norm is replaced by an arbitrary norm on ${mathbb R}^n$.
Let $x_1,ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $mu $ on ${mathbb R}^n$, and consider the random polytope $$K_N:={rm conv}{ pm x_1,ldots ,pm x_N}.$$ We provide sharp estimates for the quermass{}integrals and other geometric parameters of $K_N$ in the range $cnls Nlsexp (n)$; these complement previous results from cite{DGT1} and cite{DGT} that were given for the range $cnls Nlsexp (sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(mu )$ of $mu $ for all $1ls qls n$.
In this paper, the functional Quermassintegrals of log-concave functions in $mathbb R^n$ are discussed, we obtain the integral expression of the $i$-th functional mixed Quermassintegrals, which are similar to the integral expression of the $i$-th Quermassintegrals of convex bodies.
The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${mathbb R}^n$ asks whether for every convex body $K$ in ${mathbb R}^n$ and all $1leqslant kleqslant n$ $$Phi_{[k]}(K):={rm vol}_n(K)^{-frac{1}{n}}left (int_{G_{n,k}}{rm vol}_k(P_F(K))^{-n},d u_{n,k}(F)right )^{-frac{1}{kn}}leqslant csqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
