Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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both convex and superharmonic function on manifold concave?
M is a non-compact Rimannian manifold without boundary.
$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e.
$$
-\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M \...
- 689
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628
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Lipschitz boundary vs rectifiable curve boundary
I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...
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73
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Optimal radiating $(d{-}1)$-flats within a sphere
Permit me to revisit an earlier unresolved MO question,
"Chord arrangement that avoids confining small or large disks"
with a (very!) specific version, inspired by radiation therapy.
The main idea is ...
- 151k
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89
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Is a $CD(K,\infty)$ space a length space?
Let $(X,d)$ be a complete and separable metric space endowed with a nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray}
\mu(B(x,r))<\infty,\quad\mbox{for every }x\in X\...
- 11
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92
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rational rotation vector and closed curves
On $\mathbb{T}^n$ with a Riemannian metric, the stable norm is defined as
$$\Vert h\Vert=\inf \sum |r_i| \cdot \mathrm{length}(\sigma_i),$$
where $h\in H_1(\mathbb{T}^n,\mathbb R)$ and $\sum_i r_i\...
- 197
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197
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Area on the unit sphere swept out by big circles corresponding to a curve
For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...
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104
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Random Sequential Adsorption of Discs on a Plane - What is the best known lowerbound for the number of circles (of some radius $r$) guaranteed to fit on $[0, 1]^2$?
Imagine I perform a random sequential adsorption (RSA) simulation for circles or discs of some radius $r \leq 1$ in $[0, 1]^2$ (I am open to changing this geometry to the unit circle). As a function ...
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203
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geodesics on a Finsler space
A positive $1$-dimensional parametric integrand (short $1$-p.i.) is a continuous function on the tangent space of a manifold $F:TM\rightarrow \mathbb{R}_{\geq 0}$ that is positively homogeneous. This ...
- 1
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156
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Does homeomorphism preserves the family of cones?
Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 +...
- 165
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194
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relation with jacobifields in a small neighbourhood
hi,
I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
- 89
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289
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Dither in Leech lattice quantization!
Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the Voronoi region of the Leech lattice.
Best,...
- 197
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761
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Action of Isometries on a Line in the Plane
I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...
- 11
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695
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Minimum distance between two data sets
Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...
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2
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242
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Does uniform convergence of the metrics imply uniform convergence of the radii of the smallest balls?
Let $X$ be a countable set and $d_n,d$ locally finite metrics on $X$. Denote by $R_x^n$ (resp. $R_x$) the radius of the smallest closed ball in the metric $d_n$ (resp. $d$) about $x$ which contains at ...
- 6,034
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Question about the proof of the fact that IR is not quasi-isomtric to IR^2 [closed]
Hello.
Yesterday we proofed, that $\mathbb{R}$ is not quasi-isometric to $\mathbb{R}^2$ (both endowed with the standard Euclidean metric).
Step 1.: $\mathbb{R}$ is q.i. $\mathbb{Z}$ and $\mathbb{R}^...
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327
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Minimum cardinality of the intersection of 2D rectangles
Let $S$ be a set of 2D points $(x,y)$ with positive real coordinates, i.e. $x,y>0$. An 2D rectangle $R$ is called an ${Origin-Rectangle}$ if it is decided by the origin $(0,0)$ and another point $(...
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250
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Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces
Swenson proved in "A cut point theorem for ${\rm CAT}(0)$ groups" that a locally compact Hadamard space with a geometric action by a group $G$ admits a hyperbolic isometry (that lie in $G$).
Is it ...
- 19
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219
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Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$
Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to ...
- 499
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251
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3d width and cross section
Greetings,
We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We ...
- 3
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33
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
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72
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Reflections of Voronoi diagrams
I wonder if something similar to the following fact is known, and I would greatly appreciate any references.
Let $t_1, t_2, \ldots, t_N$ be unit vectors in $\mathbb{R}^n$.
Let $S$ denote the unit ...
- 1
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42
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Reference request: in Alexandrov geometry gradient flows preserve extremal subsets
It is mentioned in literature that in Alexandrov geometry gradient flows of semi-concave functions preserve each extremal subset.
I am looking for a proof of this fact.
- 21.8k
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37
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
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29
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Stable gap-less packing of a box with boxes
define a box packing as gap-less if
all inner boxes have disjoint interior
the sum of volumes of the inner box equals that of the outer box
the sum of the extents of the inner boxes in each principal ...
- 13.2k
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25
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Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049
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176
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How to find a configuration of lines
In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
- 61
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67
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Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 410
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21
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Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
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106
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Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$?
Any helpful answer would be appreciated. Thank you!
- 61
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Is there a log-concave distribution not spherical symmetric s.t $ \langle X, \theta \rangle$ is almost normal for all directions $\theta$?
Klartag's results indicate that for a log-concave isotropic random vector, with high probability over $\theta$, $\langle X, \theta \rangle$ is close to a normal distribution.
It is known that for the ...
- 11
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119
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Boundedness of 2 times the unit ball
Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball
$$
B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X,
$$
is it necessarily ...
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37
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Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
- 151
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78
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Coordinates of the centers of the insphere and circumsphere
Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one ...
- 1,109
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96
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When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
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231
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Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
- 1,776
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82
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On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
- 5,979
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175
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A question on Cheeger-Colding theory
I'm reading Compactification of certain Kähler manifolds with nonnegative Ricci curvature by Gang Liu recently. And I feel hard to understand a statement in the paper. Now the assumption is $(M,g)$ is ...
- 241
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89
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Practical applications of dandelin spheres
I know that dandelin spheres can be used to prove the focal properties of conic sections, but I heard that they can be used to help track the orbits of planets. All the sources I looked up only said ...
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47
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Rauch comparison theorem for $C^{1,1}$ metrics
If $g$ is a smooth riemannian metric on $M$ with nonpositive sectional curvature, the Rauch comparison theorem implies that $(M,d_g)$ is a negatively curved metric space (every point has a ...
- 151
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42
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Regularization of the Laplacian on $\mathbb{R}^d$ and approximation schemes
this question is somewhat naive, but I am trying to understand the meaning of the regularized resolvant of the Laplacian on $\mathbb{R}^d$, and how it relates to a discrete approximation. Particularly,...
- 841
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28
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Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
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41
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Trying to extend a theorem on Tiling with mutually non-congruent triangles
In the light of Cubing the cube - as 'perfectly' as possible, We try to slightly 'relax' the main theorem proved by Kupaavski, Pach and Tardos here:
https://arxiv.org/pdf/1711.04504.pdf
...
- 5,979
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77
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Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
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197
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Find which section of a convex polytope a point belongs to
Consider the convex polytope in dimension $n$ with vertex set $V$ given by the origin and the $n$ points
$$
e_i=\begin{bmatrix}0,\dots,0,\underset{i\text{-th coordinate}}{1},0,\dots,0\end{bmatrix}, i\...
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0
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56
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Seek a partition of $\mathbb R^d$
Let $c>0$ be given. I look for $n\ge 1$ and a collection of closed subsets $(F_i: 1\le i\le n)$ such that
$$\bigcup_{1\le i\le n} {\rm int}(F_i)= \mathbb R^d,$$
and for every $x\in \mathbb R^d$, ...
- 1,334
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0
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40
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What is the dual of a hyperbolic configuration of points?
Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
- 5,215
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49
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Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?
By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.
Question: For specified ...
- 5,979
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79
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On 'Width Equalizers' of planar convex regions
Definitions: The least width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C (in what follows, we refer to this quantity as simply '...
- 5,979
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67
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Comparing partitions of a given planar convex region into pieces with equal diameter and pieces of equal width
We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
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