Search Results

The second claim is false for $n=3$. Choose $\varepsilon$ small and $\delta\ll\varepsilon$. Let $A$ be the set of all points $(x,y,z)\in\mathbb R^3$ satisfying the following inequalities: $$ \begin{c …

No. There is a length metric counter-example in dimension 3. See Theorem 1 in this paper. Let me briefly explain the construction here. For a large $n$, consider a unit segment $[p_nq_n]\subset \math …

The answer to 2 is yes. Let $(X,d)$ be complete but not compact, then for some $r>0$ it has a countable r-separated subset $S=\{s_i\}:i=1,2,\dots$. Let $d'$ be the maximal metric on $X$ satisfying $d' …

There is a "contraction space" which is not complete. For example, consider a metric $d$ on $[1,+\infty)$ such that for $x,y\in[n,n+1]$ where $n\in\mathbb N$ one has $d(x,y)=2^{-n}|x-y|^{1/n}$ (other …

Yes. Let $A=(A_1,A_2)$, $B=(B_1,B_2)$ and $C=(C_1,C_2)$. For the triangle $A_1B_1C_1$ in the first tree, there is a "center" $M_1$ such that the (unique) geodesics $[A_1B_1]$, $[A_1C_1]$ and $[B_1C_1] …

The following 6 distances between 4 points $a,b,c,d$ can not be realized in a Euclidean space of any dimension: $d(a,b)=d(b,c)=d(a,c)=1$ and $d(a,d)=d(b,d)=d(c,d)=0.51$, although all triangle inequali …

A (Euclidean) polyhedral space is a metric space obtained by "gluing together" several (let's assume finitely many) Euclidean simplices (of varying dimensions) by identifying some faces via isometries …

I think I can prove that $diam(\tilde M)\le m\cdot diam(M)$ for any covering. Let $\tilde p,\tilde q\in\tilde M$ and $\tilde\gamma$ be a shortest path from $\tilde p$ to $\tilde q$. Denote by $p,q,\ga …

The answer to the title question is yes (well, I assume that by a "surface" you mean something reasonable, like a boundary of a convex set). Let $AB$ be the longest segment with endpoints on the surf …

I can complete Anton's plan with an additional assumption that geodesics do not branch. I also assume local compactness (otherwise there are too many technical details to deal with). More precisely, I …

This is an expansion of Anton Petrunin's comment. Let me describe how to perturb the standard metric of the plane so that the resulting metric is bi-Lipschitz to the original with Lipschitz constant …

The uniform distribution on the circle is optimal. Every probability measure on the disc can be approximated by the sum of atomic measures with equal wieghts, that is, by measures of the form $\frac1 …

We need to check $\eta(u,v)+\eta(v,w)\ge\eta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$. First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an …

The exclusion of horizontal lines from the assumption of the theorem does not make a big difference. If all non-horizontal lines are sent to lines, then all non-horizontal (2-dimensional) planes are …

The shadow boundary can be any $C^\infty$ curve with (quadratically) strictly convex projection to the $xy$-plane. For simplicity, let me stick to the case when the projection is a circle. So conside …