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There is no hope of making any such metric continuous with respect to $\theta$ at $s=0$ without violating the triangle inequality even in the plane. Consider parallel lines $L_1$ and $L_2$ which are s …

Let $(M,d)$ and $(N,\rho)$ be metric spaces. A function $f: M \to N$ is Lipschitz if there exists some constant $\kappa \geq 0$ so that $\rho(f(x),f(y))$ is smaller than $\kappa d(x,y)$ for all points …

This question is inspired by a recent talk by Matt Kahle on random geometric complexes. Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean spa …

Question For which pairs $M,N$ of compact metric spaces does there exist a metric space $K$ along with isometric embeddings $i:M \to K$ and $j:N \to K$ so that the Hausdorff distance between $i(M) …

Nice question! I ran into a similar problem a few years ago -- even for dimension $10$, the rejection method was annoyingly slow. One of the problems is that such questions straddle at least three hug …

Let me complement (and compliment?) Igor's answer by providing an explicit definition of bounded geometry from the work of Cheeger and Gromov. A Riemannian manifold $(M,g)$ has $C^k$-bounded geometr …

I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this wo …

Background: Let $X$ be a smooth, compact Riemannian submanifold of euclidean space $\mathbb{R}^n$. H Weyl's tube formula asserts that for sufficiently small $t > 0$, the volume $V(X;t)$ of the radius- …

The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around …

Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to …

When dealing with the problem of extending a Lipschitz function $f:A \to Y$ between metric spaces across an inclusion $A \hookrightarrow X$, one often imposes (conditions which imply) the following pr …

There's a complete classification that has been known since 1967: look past the Springer paywall here in the paper Branko Grünbaum, L. M. Kelly. Metrically homogeneous sets, Israel Journal of Mathema …

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be $K$-biLi …

Consider two finite point sets $P$ and $Q$ from (the same) Euclidean space, assume that they have the same cardinality $n$ and fix a bijection $\phi:P\to Q$. Define an undirected bipartite graph $G_\p …

Your setup defines set-valued dynamics on $X$. More precisely, you have $$X \stackrel{p}{\leftarrow} \overline{G} \stackrel{q}{\rightarrow} X$$ where $p$ and $q$ are the obvious projection maps. The s …