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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 …

In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia …

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

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\ …

$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$. Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that $d(x,y)=d_X(x, …

Under one additional condition, the answer to this problem is affirmative. The proof involves the following implication of the Affine Desargues Axiom: The Affine Moufang Axiom: for every parallel li …

I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear spa …

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 …

The affirmative answer to this problem follows from a general result on extension of graph metrics. In the following definitions, the unordered pair $\{x,y\}$ of two elements $x,y$ is denoted by $xy$. …

Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb …

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 …

After a long search, I have finally found an existing well-known geometric tool that does exactly what is required: it compares distances without expressing them in real numbers. This measuring instru …

I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$: Theorem. Let $n$ be positive integer number. A metric space $X$ is isom …

For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$. Definition: A metric space $(X,d) …

A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., geo …