We show that for p ≥ 1, the p-th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak p-th moment provided that 2q-th and q-th integral moments of these variables are comparable for all q ≥ 2. The latest condition turns out to be necessary in the i.i.d. case.

Weak and strong moments of ℓ r -norms of log-concave vectors * Rafa l Lata la and Marta Strzelecka revised version Abstract We show that for p ≥ 1 and r ≥ 1 the p-th moment of the ℓ r -norm of a logconcave random vector is comparable to the sum of the first moment and the weak p-th moment up to a constant proportional to r. This extends the previous result of Paouris concerning Euclidean norms.

We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails.

We prove estimates for E}X : ℓ n p 1 Ñ ℓ m q } for p, q ě 2 and any random matrix X having the entries of the form aijYij, where Y " pYij q1ďiďm,1ďjďn has i.i.d. isotropic log-concave rows. This generalises the result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide the analogue bound for mˆn random matrices, which entries form an unconditional vector in R mn . We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.

For $$m,n\in \mathbb {N}$$ m , n ∈ N , let $$X=(X_{ij})_{i\le m,j\le n}$$ X = ( X ij ) i ≤ m , j ≤ n be a random matrix, $$A=(a_{ij})_{i\le m,j\le n}$$ A = ( a ij ) i ≤ m , j ≤ n a real deterministic matrix, and $$X_A=(a_{ij}X_{ij})_{i\le m,j\le n}$$ X A = ( a ij X ij ) i ≤ m , j ≤ n the corresponding structured random matrix. We study the expected operator norm of $$X_A$$ X A considered as a random operator between $$\ell _p^n$$ ℓ p n and $$\ell _q^m$$ ℓ q m for $$1\le p,q \le \infty $$ 1 ≤ p , q ≤ ∞ . We prove optimal bounds up to logarithmic terms when the underlying random matrix X has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero $$\psi _r$$ ψ r ($$r\in (0,2]$$ r ∈ ( 0 , 2 ] ) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products $$A\circ A$$ A ∘ A and $$(A\circ A)^T$$ ( A ∘ A ) T .

We give a solution to the isoperimetric problem for the exponential measure on the plane with the 1 -metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the 1 -metric or its complement) has the smallest boundary measure. The proof is based on a symmetrisation (along the sections of equal 1 -distance from the origin).

No abstract

We give a solution to the isoperimetric problem for the exponential measure on the plane with the 1 -metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the 1 -metric or its complement) has the smallest boundary measure. The proof is based on a symmetrisation (along the sections of equal 1 -distance from the origin).