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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What is the best way to subdivide a simplex?
Let $\Delta^k$ be the $k$-simplex, embedded in $\mathbb{R}^{k+1}$ in the usual way so that all edges have length $\sqrt{2}$. For $k\leq 2$, there are obvious ways to subdivide $\Delta^k$ into $2^k$ ...
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49
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Transport map to lower dimension?
Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
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1
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Sequence of 2-cylinders converging to a segment in the Gromov-Hausdorff metric
Let $\{C_i\}_{i=1}^\infty$ be a sequence of (compact) 2-dimensional cylinders with smooth Riemannian metrics with Gauss curvature at least $-1$ and geodesically convex boundary (equivalently, the ...
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Understanding Gromov's metric measure space
Sorry for organized the question badly. My supervisor forced me to read chapter $3\frac 12$ of the reputed book Metric structures for riemannian and non-riemannian spaces written by Mikhail Gromov, ...
- 21
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4
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763
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Parametrizing the realization space of a polyhedron by its edges
I alluded to this here, but at that point I hadn't really done enough work to know what I wanted to ask.
Call a polyhedron "trihedral" if three faces meet at each vertex. Each of the F faces can be ...
- 13.6k
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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 $m<n$ ...
- 63.9k
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Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
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The "stained glass window problem": Draw many random chords in a circle; which kind of polygon ($3$-gon, $4$-gon, etc.) occupies the most total area?
Draw $n$ random chords in a circle, where each chord connects two independent uniformly random points on the circle.
As $n\to\infty$, which kind of polygon (triangle, quadrilateral, pentagon, etc.) ...
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Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 128k
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Is there a continuous partition of space into circles?
Question 1. Is there a continuous partition of space $\mathbb{R}^3$ into circles?
I strongly suspect not.
It is well-known by diverse arguments that space can be partitioned into circles. There is an ...
- 12.3k
10
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2
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255
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Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
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How does a conformal transformation affect the frame bundle metric of that manifold?
Suppose I have a metric $g_{\mu\nu}$ over an n-dimensional smooth orientable Riemannian manifold $M$. We then utilize Cartans repere mobile (moving frames) to define oriented orthonormal frames $e^{a}=...
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What happens if you strip everything but the “between” relation in metric spaces
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously,...
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Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
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Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
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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
22
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1
answer
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Random distance matrices
My question is motivated by the following recent paper:
Gadgil, Siddhartha; Krishnapur, Manjunath, Lipschitz correspondence between metric measure spaces and random distance matrices, Int. Math. Res. ...
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1
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Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
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Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
4
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3
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927
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Lower bound for the normal injectivity radius
Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a ...
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267
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Characterizing the D4 lattice as a sphere packing
Suppose I pack spheres in $\mathbb{R}^4$ in such a way that each touches 24 others. (All spheres in my question are assumed to have equal radius and be non-overlapping.) Does this packing ...
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0
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30
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Grid point density with the most (Demaine) neighbors
Here is a random distribution of points on a $12\cdot 12$ grid illustrating Demaine neighbors (as you can see, it can happen that a horizontal or vertical has no points at all - that doesn't cause ...
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Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
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1
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400
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Finitely generated groups with Hölder-exotic space of ends?
The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...
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Is this monoid generated from lines in $\mathbb{R}^d$ cancellative and torsion-free?
Let $L^d = \{\mathbb{R}v \ : \ v \in \mathbb{R}^d \setminus \{0\}\}$ denote the set of lines through the origin of the real vector space $\mathbb{R}^d$. I am interested in a commutative monoid ...
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Is it necessarily true that the maximal section of a centrally symmetric convex body is always bigger than its minimal projection?
I hope everyone is doing well.
Let $K \subset \mathbb{R}^n$ be a centrally symmetric convex body $(K = -K)$. Denote by $K \mid H$ the orthogonal projection of $K$ onto $H$, where $H$ is an $n - 1$ ...
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1
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Continuity of minimizers to distance function from point to convex set
Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...
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How to eliminate angle in a Glissette equation of carried point of a line sliding along two lines not at right angles
Glissettes are the curves traced out by a point carried by a curve, which is made to slide between given points or curves. My problem specifically include a line which slides between two fixed lines (...
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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 $...
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Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide ...
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Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
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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}...
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Collapse of Moebius bands with bounded below Gauss curvature and convex boundary
Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the ...
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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 ...
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On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?
Suppose that $X$ is an $n$-dimensional topological manifold that is also metrizable, and hence equipped with some metric that induces the topology.
For every point $x \in X$, let $B_\delta(x)$ be the ...
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3
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Optimal 8-vertex isoperimetric polyhedron?
I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
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Question on constraints
Does there exist any set of 6 real numbers $a_{ij}$ for $1 \leq i < j \leq 4$ satisfying the following conditions:
\begin{aligned}
&0 \le a_{ij} \le \pi, \\
&a_{ik} + a_{jk} > a_{ij}, \\
...
- 60.5k
5
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1
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Maximal symmetries of complete metrics on manifolds
Let $M$ be a connected and second-countable manifold, and $d,d'$ be $2$ complete compatible metrics on it. The isometry group of $(M,d)$, denoted as $\mathrm{Iso}(d)$, is equipped with the compact-...
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Geometric interpretation of $BN$-pairs
My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).
$[...
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114
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Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 188k
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A generalization of the law of tangents
The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Let $a$, $b$, and $c$ be the lengths of the three ...
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Obtaining the geodesic extension property by embedding in a larger space
Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
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Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
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316
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Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 5,409
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0
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103
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Low dimensional symmetric Euclidean immersions of the Klein quartic
I was wondering what the lowest Euclidean dimension the Klein quartic has a faithful symmetric immersion, that is an immersion such that all of its automorphisms* are distinct symmetries of the ...
- 2,370
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0
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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!
- 63.9k
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1
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On some infinite planar arrangements with triangles
Background: Given a convex region C. One can define a graph corresponding to a planar arrangement of non overlapping congruent copies of C - each unit C is a node and an edge connects it to another ...
CommunityBot
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Question on the exact largest minimum angle
Could anyone help find the EXACT largest minimum angle between any pair of lines among 5 lines passing through the origin in $\mathbb{R}^3$? Additionally, what is the exact largest minimum angle ...
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8
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Convex hull in CAT(0)
Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset.
Is it true that convex hull of $K$ is compact?
Comments:
Convex hull of $K$ = intersection of all closed convex sets ...
3
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1
answer
239
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The realization space of non-convex polyhedra - What is known?
The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
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