**1**

vote

**0**answers

30 views

### An optimality condition for the corners of convex polytopes?

Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$).
Let's ...

**1**

vote

**0**answers

38 views

### Of the standard distance metrics, which ones can/cannot be embedded in Euclidean space?

Given the discussion from:
Representability of finite metric spaces
it appears that a 1974 paper by Morgan gives the criteria for when a distance metric can be embedded in Euclidean space. My first ...

**4**

votes

**2**answers

126 views

### Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?

If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...

**4**

votes

**1**answer

47 views

### Quantitative version of the splitting theorem

The classical splitting theorem (Toponogov, Milka) says that if a smooth complete Riemannian manifold (more generally, Alexandrov space) $M^n$ of non-negative sectional curvature contains a line (i.e. ...

**3**

votes

**0**answers

113 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...

**2**

votes

**1**answer

128 views

### How often can a single length occur as a boundary distance?

Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$?
We can make any assumptions about the ...

**46**

votes

**1**answer

543 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...

**0**

votes

**1**answer

152 views

### Is the map $\exp_x(\nabla_x \sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification.
Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...

**11**

votes

**0**answers

63 views

### A kaleidoscopic coloring of the plane

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 ...

**21**

votes

**1**answer

4k views

### L1 distance between gaussian measures

L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...

**6**

votes

**1**answer

113 views

### Existence of a measurable map between metric spaces

Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
Is ...

**3**

votes

**1**answer

42 views

### Is the boundary of Alexandrov space again an Alexandrov space?

Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the ...

**68**

votes

**11**answers

9k views

### Is it possible to capture a sphere in a knot?

You and I decide to play a game:
To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...

**6**

votes

**2**answers

85 views

### Equality of triangles in normed spaces

Hilbert space satisfies the following condition: if two triangles $\triangle ABC$, $\triangle A_1B_1C_1$ have equal sides lengths: $|AB|=|A_1B_1|$, $|BC|=|B_1C_1|$, $|AC|=|A_1C_1|$ they also have ...

**2**

votes

**2**answers

996 views

### The Area of Spherical Polygons

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can ...

**8**

votes

**1**answer

264 views

### Embedding Z into Z^2 with large distortion

Is it possible to find a 2-way infinite (self-avoiding) path $\{x_i\}_{i\in \mathbb Z}$ in the standard Cayley graph of $\mathbb Z^2$, i.e. the square grid, such that the distance between $x_i$ and ...

**41**

votes

**7**answers

3k views

### Is there an algebraic approach to metric spaces?

It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there ...

**9**

votes

**1**answer

541 views

### Ordered geometries from convex subsets of the plane

Motivation
In the Klein disk model of the hyperbolic plane, the points are the interior of the disk, and the lines in $H^2$ correspond to lines intersecting the interior.
Similarly, the Euclidean ...

**5**

votes

**1**answer

111 views

### If all balls around fixed basepoints are isometric, are the spaces as well (length spaces)?

Let $X$ and $Y$ be two complete proper length spaces, $x \in X$ and $y \in Y$.
Assume for every $r>0$ the closed balls $\overline{B_r(x)}$ and $\overline{B_r(y)}$ are isometric.
Does there exist ...

**6**

votes

**2**answers

811 views

### Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...

**7**

votes

**1**answer

249 views

### Area of square to wrap a torus

The Nash-Kuiper
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
has recently been spectacularly visualized by the
Hevea Project.
This suggests two questions.
Q1. What is the area of ...

**2**

votes

**1**answer

63 views

### A property of geodesic triangles in Alexandrov spaces

Let $X$ be an $n$-dimensional Alexandrov space with curvature at least -1. Assume that at every point it has an $(n,\delta)$-strainer of length $\mu$, where $\delta$ and $\mu$ are independent of a ...

**15**

votes

**6**answers

2k views

### Almost orthogonal vectors

This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or ...

**28**

votes

**5**answers

2k views

### Is Lebesgue's “universal covering” problem still open?

The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ...

**41**

votes

**2**answers

2k views

### Randall Munroe's Lost Immortals

In Randall Munroe's book What If?, the "Lost Immortals" question asks:
If two immortal people were placed on opposite sides of an uninhabited Earthlike planet, how long would it take them to find ...

**6**

votes

**0**answers

95 views

### Bang's open question strengthening Tarski's planks problem

Tarski's Planks problem,
solved by Thøger Bang in 1951, says (in a simplified $\mathbb{R}^2$ version) that it requires
"planks" (parallel strips) of total width $\ge d$ in order to completely cover
a ...

**2**

votes

**0**answers

81 views

### What is the metric on the Fuchsian model? [closed]

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...

**4**

votes

**1**answer

85 views

### A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius

Does there exist a function $\tau(\varepsilon)=\tau(\varepsilon,n,K,\mu)$ such that $\lim_{\varepsilon\to +0}\tau(\varepsilon)=0$ and for any $n$-dimensional complete Riemannian manifold $M^n$ with ...

**7**

votes

**2**answers

238 views

### Isometric imbedding of a sphere with positively curved metric

QUESTION. Given a Riemannian metric on the sphere $S^n$ with positive sectional survature. Can it be isometrically imbedded into $\mathbb{R}^{n+1}$ (of any class of regularity) as a boundary of a ...

**19**

votes

**5**answers

2k views

### Unexpected applications of Dvoretzky's theorem

Dvoretzky's theorem is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem ...

**1**

vote

**1**answer

27 views

### Multiplicity of a subcovering in spaces of given Hausdorff dimension

Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...

**4**

votes

**0**answers

73 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

**1**

vote

**0**answers

52 views

### Lebesgue differentiation theorem holds on locally doubling space?

It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...

**4**

votes

**1**answer

94 views

### local quasi geodesics in hyperbolic spaces

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too.
We have the following two well-known Theorems:
T1) For all $\delta > 0, ...

**5**

votes

**1**answer

235 views

### Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

**6**

votes

**1**answer

179 views

### Hilbert metric and cross-ratio of points on simplices

Background and motivation:
Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (those for which $\sum ...

**1**

vote

**1**answer

77 views

### Pairs of rays in euclidean buildings

In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...

**53**

votes

**11**answers

6k views

### Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

**2**

votes

**0**answers

49 views

### construction of grassmannian manifolds as collection of subspaces of Euclidean space

The grassmannian $G_k(\mathbb{R}^n)$ is the collection of all $k$-dimensional linear subspaces of $\mathbb{R}^n$ equipped with the quotient topology. The cohomology ring of $G_k(\mathbb{R}^n)$ has ...

**3**

votes

**1**answer

192 views

### Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...

**22**

votes

**2**answers

1k views

### Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...

**17**

votes

**4**answers

726 views

### Is Monsky's theorem depending on axiom of choice ?

The extension of the 2-adic valuation to the reals used in the usual proof uses clearly AC. But is this really necessary ? After all, given a equidissection in $n$ triangles, it is finite, so it ...

**21**

votes

**3**answers

2k views

### Distributing points evenly on a sphere

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible.
I have found some related questions on stackoverflow but ...

**16**

votes

**5**answers

2k views

### How Does One Find the “Loneliest Person on the Planet”?

I'm looking for the algorithm that efficiently locates the "Loneliest Person on the Planet", where "loneliest" is defined as:
Maximum minimum distance to another person -- that is, the person for ...

**4**

votes

**1**answer

150 views

### when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...

**2**

votes

**0**answers

42 views

### $|\exp_p(x)\exp_q(T(x))|$ controlled by $|pq|?$ $T$ is parallel transportation in Alexandrov space

Let $M$ be an Alexandrov space with $sec \geqslant 0$. Let $p$ and $q$ be points of shortest path $\gamma$ in $M$, that are not end point. Then the tangent cone can split as $C_p=L_p \times ...

**11**

votes

**3**answers

224 views

### Curvature of a finite metric space

I am sorry to ask a very vague question, but:
What are good ways to define the curvature of a finite metric space?
The best way I can think of is: the curvature of a finite metric space $M$
is ...

**5**

votes

**1**answer

64 views

### Volume satisfying inequality constraints (simplex subset)

Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...

**1**

vote

**1**answer

78 views

### 4D Duoprisms based on nonconvex polygons

A duoprism is a polytope
that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$).
Four-dimensional duoprisms in particular have been studied:
$$P \times Q = \{ ...

**15**

votes

**1**answer

482 views

### Lower-Hölder embeddings of the sphere

My question is very simple:
Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that
$$
|f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r,
$$
for ...