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Is the following metric characterization of the real line true (and known)? A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real …

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap …

I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book "G …

Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n: …

Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??) implies …

The question was motivated by this question of Anton Petrunin. By a metric continuum we understand a connected compact metric space. Let $p$ be a positive real number. A metric continuum $X$ is call …

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, …

It seems that the conjecture (H2) can be confirmed with help of the recent result of Hofmann and Kramer (http://arxiv.org/pdf/1301.5114.pdf) who proved that for a compact topological group $G$ and a c …

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points …

The answer to this question is positive. A required path $\gamma$ can be constructed inductively using the following Lemma. For any continuum $P\subset\mathbb R^2$, distinct points $x,y\in P$, and $\ …

I am looking for (maybe existing) name for a mathematical structure $(X,\leqslant)$ consisting of a set $X$ and a transitive relation ${\leqslant}\subseteq X^2\times X^2$ such that $xx\leqslant yz\leq …

Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space? A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder …

On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the f …

By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\t …

For a compact space $Y$ the answer is affirmative, but in general case of Polish space $Y$ it is negative. Results yielding nice selections of relations $R$ are known in Descriptive Set Theory as Un …