Related papers: Definable versions of theorems by Kirszbraun and H…

The classical Kirszbraun theorem says that all $1$-Lipschitz functions $f:A\longrightarrow \mathbb{R}^n$, $A\subset \mathbb{R}^n$, with the Euclidean metric have a $1$-Lipschitz extension to $\mathbb{R}^n$. For metric spaces $X,Y$ we say…

A direct application of Zorn's Lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$ has an extension to a Lipschitz map $\widetilde f: \mathbb{Q}_p^n\to \mathbb{Q}_p^\ell$. This is analogous, but more easy,…

In this paper, we prove a definable version of Kirszbraun's theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, we prove that every definable function $f : X \times Y \to…

In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal fields $K$. This may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's extension theorem. We proceed…

Let $X,Y$ be two Hilbert spaces, $E$ a subset of $X$ and $G: E \to Y$ a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists $\widetilde{G} : X \to Y$ with $\widetilde{G}=G$ on $E$ and…

In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$.…

Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We…

The well-known Reifenberg theorem states that if a subset of $\mathbb{R}^n$ can be well approximated by $k$-planes at every point and every scale, then it is biH\"older homeomorphic to a $k$-disk. This article concerns a subset $S$ of…

Szpilrajn's Lemma entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn's Lemma to show that each partial order has a relizer. Since then, many authors utilize Szpilrajn's Theorem and the Well-ordering…

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or subanalytic sets. A {\em monotone map} is a multi-dimensional generalization of a usual univariate monotone function, while the…

The following theorem is proved: Let M be a locally Lipschitz hypersurface in C^n with one-sided extension property at each point (e.g., without analytic discs). Let S be a closed subset of M and f : M \ S ---> C^m \ E is a CR-mapping of…

We show that for arbitrary linearly ordered set $X$ any bounded family of (not necessarily, continuous) real valued functions on $X$ with bounded total variation does not contain independent sequences. We obtain generalized Helly's…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

We study the problem of extending any order-preserving Lipschitz function that maps a subset of a partially ordered Hilbert space X into a Hadamard poset Y without increasing its Lipschitz constant and preserving its monotonicity. This sort…

Garret Birkhoff's HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra $\mathbf{B}$ satisfies…

We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are…

Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a…

We identify a canonical structure J associated to any first-order theory, the {\it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple…

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We…