Given a normalized Orlicz function M we provide an easy formula for a distribution such that, if X is a random variable distributed accordingly and X₁,...,Xₙ are independent copies of X, then $1/C_p{\left|\right|x\left|\right|}_M\le \left|\right|\left(x_i{X}_i\right){ⁿ}_{i=1}{\left|\right|}_p\le C_p{\left|\right|x\left|\right|}_M$, where $C_p$ is a positive constant depending only on p. In case p = 2 we need the function t ↦ tM’(t) - M(t) to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into L₁[0,1]. We also provide a general result replacing the ${\ell }_p$-norm by an arbitrary N-norm. This...

In [K-S 1] it was shown that $Av{e}_{\pi }({\sum }_{i=1}^n|x_i{a}_{\pi \left(i\right)}{{|}^2)}^{1/2}$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1,...,a_n$ so that the above expression is equivalent to a given Orlicz norm.

We compare various constructions of random proportional quotients of $l{₁}^m$ (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.

This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^p$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^p$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques...

Let 1 ≤ p < 2 and let $L_p=L_p[0,1]$ be the classical $L_p$-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f\in L_p$ spans in $L_p$ a subspace isomorphic to some Orlicz sequence space $l_M$. We give precise connections between M and f and establish conditions under which the distribution of a random variable $f\in L_p$ whose independent copies span $l_M$ in $L_p$ is essentially unique.

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_N$-spaces needed to represent (up to a constant $C>1$) the $M_N$-version of the $n$-dimensional operator Hilbert space $O{H}_n$ as a direct sum of copies of $M_N$. We show that, when $C$ is close to 1, this multiplicity grows as $exp\beta n{N}^2$ for...

We survey some questions on Rademacher series in both Banach and quasi-Banach spaces which have been the subject of extensive research from the time of Orlicz to the present day.

We show that, given an n-dimensional normed space X, a sequence of $N={(8/\epsilon )}^{2n}$ independent random vectors ${\left(X_i\right)}_{i=1}^N$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $\Gamma :ℝ\to {ℝ}^N$ defined by $\Gamma x={(⟨x,X_i⟩)}_{i=1}^N$ embeds X in ${\ell }_{\infty }^N$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ${\ell }_{\infty }^N$ with asymptotically best possible relation between N, n, and ε.

We present a general result on regularization of an arbitrary convex body (and more generally a star body), which gives and extends global forms of a number of well known local facts, like the low M*-estimates, large Euclidean sections of finite volume-ratio spaces and others.

We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty }$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by ℓ₁-minimization under the optimal condition m ≥ csln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the ℓ₁-norm and the outer norm depends on probability distributions.

We show that a Banach space X has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if X has nontrivial type. If for every Radon probability on X, there is an operator from an $L_p$ space into X whose range has probability one, then X is a quotient of an $L_p$ space. This extends a theorem of Sato’s which dealt with the case p = 2. In any infinite-dimensional Banach space X there is a compact set K so that for...

We formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.