The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f:S^k\to {ℝ}^k$ there exists a point $x\in S^k$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^k$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of $in{f}_x\left|\right|f\left(x\right)-f(-x)\left|\right|$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

In this paper we prove a representation result for essentially bounded multivalued martingales with nonempty closed convex and bounded values in a real separable Banach space. Then we turn our attention to the interplay between multimeasures and multivalued Riesz representations. Finally, we give the multivalued Radon-Nikodym property.

We prove that if $r_i$ is the Rademacher system of functions then ${\left(ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right){\parallel }^{2}dt\right)}^{1/2}\le \surd 2ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right)\parallel dt$ for any sequence of vectors $x_i$ in any normed linear space F.

In this note we investigate the relationship between the convergence of the sequence $\left\{S_n\right\}$ of sums of independent random elements of the form $S_n={\sum }_{i=1}^n{\epsilon }_i{x}_i$ (where ${\epsilon }_i$ takes the values $\pm 1$ with the same probability and $x_i$ belongs to a real Banach space $X$ for each $i\in \mathbb{N}$) and the existence of certain weakly unconditionally Cauchy subseries of ${\sum }_{n=1}^{\infty }{x}_n$.

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.