# Tagged Questions

**2**

votes

**0**answers

16 views

### Carving a rectilinear polygon

In this question, carving a polygon $P$ means removing an axis-parallel rectangle adjacent to the boundary of $P$. Carving $P$ might break it into two or more polygons.
You are given a square $P$. ...

**2**

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**0**answers

51 views

### Christoffel symbols of a moduli of smooth 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

42 views

### Example of a compact geodesic space, which is not doubling [on hold]

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

**3**

votes

**1**answer

65 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 $x \in \mathbb{R}^2$ ...

**5**

votes

**2**answers

143 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 ...

**5**

votes

**2**answers

314 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

144 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 ...

**2**

votes

**0**answers

90 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 ...

**0**

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**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 ...

**2**

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**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 $\gamma:[a,b]\to\mathbb R^n$ ...

**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 ...

**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 ...

**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 ...

**11**

votes

**1**answer

209 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 ...

**0**

votes

**0**answers

60 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 ...

**8**

votes

**2**answers

449 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 ...

**1**

vote

**2**answers

126 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 $Y$ ...

**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**

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**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 $x\in X$, ...

**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 ...

**13**

votes

**1**answer

355 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 ...

**4**

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**0**answers

83 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 ...

**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 ...

**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

118 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 $\mathbb{R}$, ...

**2**

votes

**1**answer

89 views

### What is the maximal diameter of a cell in a particular partition of the simplex?

Consider a standard simplex with points $(p_1, \dots, p_n)$, $p_i \ge 0$, and $\sum_i p_i = 1$. Fix a set $\{q_k\}_{k=1}^K$ with $0\leq q_k \leq \infty$ and $i,j\in\{1, \dots, n\}$. Partition it via ...

**5**

votes

**1**answer

182 views

### Compact Eucledean hypersurfaces with “almost” constant H_k curvature

Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...

**6**

votes

**1**answer

283 views

### Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space ...

**1**

vote

**0**answers

56 views

### Lipschitzian extension of mapping between Alexandrov spaces

Let $X$ be an $n-$dim (compact, if needed) Alexandrov space with curvature $\geq -k$, with $k\geq0$, and let $Y$ be an Alexandrov space with curvature $\leq0$ globally. Given any bounded nonempty ...

**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 ...

**8**

votes

**0**answers

189 views

### Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof.
Question: Let $M$ be a ...

**10**

votes

**1**answer

215 views

### Optimization of points on a plane

Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...

**3**

votes

**1**answer

137 views

### Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem:
Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...

**3**

votes

**0**answers

69 views

### Is the doubling dimension invariant under Möbius maps?

Is the doubling dimension of a metric space $(X,d)$ invariant under maps $f:(X,d) \to (X,d')$, which are the identity on $X$ and which map the metric $d$ to a metric $d'$ which has the same cross ...

**6**

votes

**0**answers

207 views

### Axiom of choice and a set in the plane that intersects every line in two points

In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects ...

**1**

vote

**0**answers

45 views

### Starlike curve tangent condition

Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$
denote the acute angle which the ...

**0**

votes

**0**answers

81 views

### Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]:
$$\begin{align}
\theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\
& = \frac{1}{2} \| ...

**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 ...

**0**

votes

**0**answers

55 views

### What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?

I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...

**6**

votes

**1**answer

87 views

### metric condition forcing convex position

Let $A_1,A_2,\ldots, A_n$ be distinct in the plane. For every $1\le i \le n$, let
$S_i=\sum\limits_{j=1}^n d(A_i,A_j)$ be the sum of distances from $A_i$ to all the other points.
Assume that ...