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

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3
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1answer
128 views

Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where $$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...
12
votes
2answers
255 views

Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity. For instance, a subset of ...
7
votes
1answer
198 views

Partitioning a convex object without harming existing convex subsets

$C$ is a convex planar figure and $C_1,\dots,C_n$ are pairwise-disjoint convex subsets of $C$, like this: A convex-preserving partition of $C$ is a partition $C=E_1\cup\dots\cup E_N$, , such that ...
10
votes
0answers
153 views

Electrons on a pancake ellipsoid

The problems of minimizing the potential energy of electrons on a sphere, or maximizing the smallest distance between the electrons, have been well-studied. E.g., see the earlier MO question ...
12
votes
1answer
238 views

Covering the unit sphere by open hemispheres

Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...
1
vote
0answers
56 views

Comparison theorem for Lambert quadrilateral

A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle. If $AOBF$ is a Lambert ...
0
votes
0answers
35 views

Minimize a function to learn a mapping

I have two questions. I want to learn a mapping $M$ that minimizes the right-hand side of the following equation: $E =\frac{1}{N} \sum\limits_{i=1}^N \bigg(\sum\limits_{j=1}^K \alpha_i - ...
5
votes
0answers
166 views

Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?

I am interested in the Hausdorff dimension of the Apollonian circle packing. There seem to be two numerical calculations of the value: 1.305686729(10) from P.B ...
-1
votes
1answer
111 views

Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$. More precisely: I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some ...
1
vote
0answers
22 views

Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...
4
votes
0answers
47 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 $P$. ...
3
votes
1answer
126 views

3-dimensional vectors satisfying certain equalities

Question: Are there 5 distinct vectors $u,v,w,x,y \in \mathbb{R}^3$, all on the unit sphere (i.e. $||u||=||v||=||w||=||x||=||y||=1$), such that: $||u+v+x||=||u+v+y||=||u+w+x||=||u+w+y||=1$ ? Also, ...
2
votes
0answers
57 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} ...
5
votes
1answer
116 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
2answers
184 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
2answers
350 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
2answers
159 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
0answers
162 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
votes
0answers
65 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
0answers
101 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
votes
0answers
51 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
1answer
266 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
1answer
86 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
0answers
82 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
1answer
227 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
0answers
27 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
1answer
94 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
0answers
63 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
2answers
514 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
2answers
134 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 $Y$ ...
4
votes
1answer
129 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
0answers
88 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 ...
15
votes
1answer
519 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
votes
0answers
85 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, ...
7
votes
0answers
168 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
2answers
905 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
2answers
187 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 ...
9
votes
0answers
217 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
0answers
62 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
2answers
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
0answers
78 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
1answer
149 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
1answer
115 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
1answer
93 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 ...
6
votes
1answer
192 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
1answer
289 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
0answers
58 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
0answers
281 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 ...
14
votes
1answer
328 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
1answer
227 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 ...