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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]).

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.

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${ℂ}^n$ of codimension 1, $vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}\left(H\cap D^n\right)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)$. The lower bound is attained if and only if H is orthogonal to the versor $e_j$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_j+\sigma e_k$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${ℂ}^n$ with ${ℝ}^{2n}$; by $vo{l}_k\left(·\right)$ we denote the usual k-dimensional volume in ${ℝ}^{2n}$. The result is a complex counterpart of Ball’s [B1] result for...

A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have $\gamma ₙ\left(sK\right)\le {\left(2s\right)}^{w²/4}\gamma ₙ\left(K\right)$ for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in ${ℝ}^k$. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.

For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = μ∘λ: λ ∈ , which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for which ℳ(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ₁, Θ₂ independent of X, X’ there exists a random variable Θ independent of X such...