Related papers: The Banach-Tarski Paradox

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order…

A classical consequence of the John Ellipsoid Theorem is the upper bound $\sqrt{n}$ on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathbb{R}^n$. Equality is attained for the parallelotope and the…

We provide the proof of a previously announced result that resolves the following problem posed by A.~A.~Kirillov. Let $T$ be a presentation of a group $\mathcal{G}$ by bounded linear operators in a Banach space $G$ and $E\subset G$ be a…

Extending the classical Levy-Steinitz rearrangement theorem, which in turn extended Riemann's theorem, Banaszczyk proved in 1990/93 that a metrizable, locally convex space is nuclear if and only if the domain of sums of every convergent…

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we…

This paper gives a counterexample to the impossibility, by G\"odel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent.…

We describe (infinite-dimensional) irreducible representations of the crossed product C$^*$-algebra associated with a topological dynamical system (based on $Z$) and we show that their restrictions to the underling $\ell^1$-Banach…

In the first part, after showing that the most natural approach to define an order on sets of conformal classes fails, we define a nontrivial order $\leq_2$ on the set of conformal classes of compact Cauchy slabs with fixed past boundary…

We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 cross A and 3 cross B then there is a one-to-one correspondence between A and B. The first such proof, due to…

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired…

We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal…

We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…

We single out and study a natural class of Banach spaces -- almost square Banach spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i-y\|$ is almost…

We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance…

We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements…

The 1-2-3 Conjecture, introduced by Karo\'nski, {\L}uczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph $G$ different from $K_2$, we can turn $G$ into a locally irregular multigraph $M(G)$,…

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~G\l{}\k{a}b, E.~Jab\l{}o\'nska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of…

We prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls.

In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wo\`{z}niakowski obtained an…