Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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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} ...
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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
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1answer
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
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0answers
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
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2answers
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
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2answers
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
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1answer
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
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2answers
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 ...
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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
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1answer
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
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0answers
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
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1answer
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
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1answer
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 ...
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0answers
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
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0answers
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
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1answer
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
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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
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1answer
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
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2answers
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
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1answer
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
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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
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1answer
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
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1answer
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 ...
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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
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1answer
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 ...
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2answers
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
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3answers
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
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2answers
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
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0answers
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
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2answers
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
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2answers
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 ...
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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
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1answer
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 ...
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0answers
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 ...
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0answers
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
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0answers
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
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1answer
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 ...
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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
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2answers
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
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2answers
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
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0answers
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
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0answers
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
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1answer
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
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2answers
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 ...
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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 ...
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0answers
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
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2answers
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
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0answers
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} ...
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1answer
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
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1answer
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 $\mathbb{R}$, ...