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}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.

Geometry of random sections of isotropic convex bodies

Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability greater than $1-exp (-sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(mu )$ of an isotropic log-concave probability measure $mu $ on ${mathbb R}^n$. For every $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. We also give bounds on the diameter of random projections of $Z_q(mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${mathbb R}^n$ then for a random subspace $F$ of dimension $(log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(log^2n)$.

Asymptotic shape of the convex hull of isotropic log-concave random vectors

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 $1ls qls n$.

Estimates for measures of lower dimensional sections of convex bodies

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${mathbb R}^n$ with $0in {rm int}(K)$ and if $mu $ is a measure on ${mathbb R}^n$ with a locally integrable non-negative density $g$ on ${mathbb R}^n$, then begin{equation*}mu (K)leq left (csqrt{n-k}right )^kmax_{Fin G_{n,n-k}}mu (Kcap F)cdot |K|^{frac{k}{n}}end{equation*} for every $1leq kleq n-1$. Also, if $mu $ is even and log-concave, and if $K$ is a symmetric convex body in ${mathbb R}^n$ and $D$ is a compact subset of ${mathbb R}^n$ such that $mu (Kcap F)leq mu (Dcap F)$ for all $Fin G_{n,n-k}$, then begin{equation*}mu (K)leq left (ckL_{n-k}right )^{k}mu (D),end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.