**2**

votes

**0**answers

20 views

### christoffel symbols of a moduli of smmoth curves

The Setting:
Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation}
<f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx
\end{equation}
...

**0**

votes

**0**answers

28 views

### Example of a compact geodesic space, which is not doubling

Are all compact geodesic spaces doubling? If not, could you give an example?

**3**

votes

**1**answer

60 views

### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...

**9**

votes

**0**answers

290 views

### diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...

**4**

votes

**2**answers

129 views

### No normal coordinates on general Finsler manifolds

I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates.
As I'm still new to non-Riemmanian Finsler geometry I don't see why ...

**24**

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 ...

**11**

votes

**1**answer

387 views

### Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...

**5**

votes

**2**answers

308 views

### What is this distance about?

For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
This question by Ritesh Ahuja (positive answered by Iosif Pinelis) says that $d$ is a metric. My ...

**4**

votes

**2**answers

142 views

### Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge ...

**1**

vote

**1**answer

251 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 ...

**2**

votes

**0**answers

80 views

### Extra large spherical joins

If $X$ and $Y$ are piecewise spherical complexes, then their spherical join $X * Y$ is CAT(1) if and only of $X$ and $Y$ are CAT(1) (see the appendix of the first Charney-Davis paper below). One of ...

**2**

votes

**1**answer

62 views

### Coordinate chart of concave functions near a regular point in Alexandrov spaces

Let $M$ be an Alexandrov space with curvature $\geqslant -1$. Then we have the following theorem which is often used to perturb a regular point to points we want.
Let $g_0$ be a ...

**4**

votes

**1**answer

143 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 ...

**0**

votes

**0**answers

62 views

### Hausdorff limits of fibers of affine maps

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and let
$$
F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m
$$
be a polynomial map. I would like to know under what conditions the preimages $F^{-1}(y)$ of ...

**3**

votes

**0**answers

92 views

### Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity.
Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...

**11**

votes

**1**answer

274 views

### Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light.
Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...

**5**

votes

**0**answers

66 views

### Regularity of the distance from the boundary in singular riemannian manifolds

I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...

**9**

votes

**1**answer

192 views

### Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...

**7**

votes

**2**answers

300 views

### Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...

**15**

votes

**1**answer

242 views

### Does a Riemannian manifold have a triangulation with quantitative bounds?

Suppose that $M$ is a closed Riemannian manifold with bounded geometry, i.e., curvature between $-1$ and $1$ and injectivity radius at least $1$. Since $M$ is a smooth manifold, it has a ...

**2**

votes

**0**answers

43 views

### Equivalence of local and global geodesics in projective spaces

Let $d:\mathbb R^n\times \mathbb R^n\to\mathbb R^n$ be a distance which induces the Euclidean topology and with which $\mathbb R^n$ is a length space.
A continuous curve $\gamma:[a,b]\to\mathbb R^n$ ...

**0**

votes

**1**answer

72 views

### Example distance metric that is not conditionally negative definite

Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...

**11**

votes

**1**answer

208 views

### Hausdorff convergence of submanifolds in Riemannian manifolds

Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...

**1**

vote

**0**answers

25 views

### Example of compact $CD(K,\infty)$ space, but doubling condition fails

It's well known that the doubling condition may not hold on $CD(K,\infty)$ space.
Can one give an example such that: $(X,d)$ is a compact metric space, $\mu$ is a Borel probability measure and ...

**3**

votes

**1**answer

80 views

### The volume of a region arising from planar linkages

Let $x_0,\dots,x_n$ be a collection of variable points in $\mathbb{R}^2$ and let $c>0$ be a fixed constant. Is there any way I could compute an upper bound of the volume of the region in ...

**1**

vote

**2**answers

101 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 ...

**38**

votes

**3**answers

3k views

### Is the “Napkin conjecture” open? (origami)

The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...

**8**

votes

**2**answers

448 views

### Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n ...

**0**

votes

**0**answers

59 views

### Joint point of coarse geometry and dynamical system?

My major interest is on dynamical systems,
but I did REU in a coarse embedding problem.
I wonder whether there's some significant connection between those two subjects.
I've tried to google for a ...

**1**

vote

**2**answers

125 views

### Antiproximanal subspace of $L_1[0,1]$

Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ ...

**8**

votes

**2**answers

389 views

### Is displacement controled by stable norm?

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a ...

**0**

votes

**0**answers

279 views

### Geodesic Digons in Reductive Spaces

Consider a naturally reductive homogeneous space $M$ of positive curvature. Is it necessarily the case that there exists a geodesic digon whose interior angles sum to less than $2\pi$? Angles are ...

**12**

votes

**1**answer

334 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any ...

**0**

votes

**0**answers

38 views

### Complex tetrad vs Real metric

I have a question on the relationship between the complex tetrad in general relativity and the metric. All the papers I've sen so far just usually state the metric and the (null) tetrad without ...

**-2**

votes

**0**answers

71 views

### Parallel bi-linear forms on $S^2$

Let $S^2$ denote the sphere endowed with its standard (curvature $k= 1$) metric $g$. How many linearly independent, parallel, symmetric bi-linear forms can we find on $S^2$ ? Clearly the metric $g$ ...

**3**

votes

**0**answers

47 views

### Does Alexandrov space satisfy a reverse doubling condition?

Let $X$ be an $n-$dim Alexandrov space with curvature $\geq k$. Does $X$ satisfy a reverse doubling condition? That is, does there exist a constant $C>1$, s.t., for any $x\in X$, ...

**8**

votes

**1**answer

223 views

### A forked plane continuum

I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...

**0**

votes

**0**answers

78 views

### Metric calculation from tetrad gives wrong answer

I'm reading the following article by Kinnersley
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
and cannot reproduce one (rather trivial) result.
On page 5 of the paper, in ...

**23**

votes

**2**answers

886 views

### How can you compute the maximum volume of an envelope(used to enclose a letter)?

It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...

**51**

votes

**2**answers

822 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 ...

**4**

votes

**0**answers

82 views

### Is positively curved Alexandrov surface isometrically embeddable in $\mathbb R^3$?

I guess it is not. The example I have in mind is: $X^2$ is the spherical suspension of a circle $S^1(t)$ of length $0<t<2\pi$. Then $X$ has constant curvature =1 except at two suspension points, ...

**6**

votes

**0**answers

155 views

### Shortest path to inspect a polyhedron

This is a variant of two as-yet unsolved MO questions cited below.
Let $P$ be a closed polyhedron in $\mathbb{R}^3$.
The task is to find a shortest path $\sigma$ on the surface of $P$ from which
all ...

**7**

votes

**1**answer

815 views

### Isoperimetry and Poincaré Inequality

What are the known relations between isoperimetric and Poincaré inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...

**4**

votes

**2**answers

175 views

### Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion
initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...

**8**

votes

**0**answers

195 views

### Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?

Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...

**1**

vote

**0**answers

57 views

### Completely incongruent box partitions

Let $B$ be a rectangular box with corners in $\mathbb{Z}^d$
and sides parallel to the axes.
A completely incongruent partition of $B$ is a partition into
$d$-dimensional boxes, each of whose integer ...

**3**

votes

**2**answers

117 views

### Sub-sector covers for disks and balls

Define an open $k$-sector of a disk as the portion between two radii separated
by an angle of $2\pi/k$, but open along the two radii (and closed along the
circle boundary).
Call a set a sub-$k$-sector ...

**3**

votes

**0**answers

73 views

### Calculating the length of curve using dyadic partition [closed]

Let $\gamma:[0,1] \rightarrow \mathbb{R}^n$ be a continuous function. The length of $\gamma$ is usually defined as
$$\sup_{0 = t_1 < t_2 < \cdots < t_n = 1} \sum_{i=1}^{n-1} ...

**1**

vote

**1**answer

123 views

### Intuition behind the “Lapse Function”

I came across the following definite of the Lapse Function:
$N=\sqrt{\frac{1}{2}g(L,\overline{L})}$
where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...

**2**

votes

**1**answer

113 views

### Under which conditions is it possible to find points with same distances under bi-Lipschitz map [closed]

Given two metric spaces $(X,d_X), (Y, d_Y)$, a bi-Lipschitz map $f:X \to Y$ and a finite set of points $\{x_1, \ldots, x_n\} \in X$. Consider in addition, that $X$ is a vector space over $\mathbb{R}$, ...