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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Can any triangle be inscribed in any convex figure?
Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the ...
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-2
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1
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Generic coordinate system representations [closed]
Please excuse the verboseness which follows, as the question is rather basic, so I would like to state it carefully so that it will not be accidentally neglected as automatically trivial. If, after my ...
- 2,481
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2
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An optimization problem for points on the sphere (master's dissertation)
First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
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8
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1
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Point cloud that maximizes the minimum pairwise distance in Euclidean space
I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize
$min_{i \neq j} |...
- 12.6k
5
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1
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Minimizing variance of distances between points when mean distance is fixed
In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d &...
- 45k
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Lattice Stick Number vs. Stick Number of Knot
Can the lattice stick number of a knot be bounded
by the stick number of the knot?
The stick number
$S(K)$ of a knot $K$ is the fewest number of segments
needed to realize it by a simple 3D polygon....
- 18.6k
1
vote
1
answer
492
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Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
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4
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literature on geometrical viewpoint on calculus of variations for physics
What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...
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2
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749
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Projective transformation between polygons.
Extending my earlier question about linear transformations, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}...
CommunityBot
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1
answer
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What is the Cheeger constant of a cubical subset of the cubic lattice?
The Cheeger constant of a finite graph measures the "bottleneckedness" of the graph, and is defined as:
$$h(G) := \min\Bigg\lbrace\frac{|\partial A|}{|A|} \Bigg| A\subset V, 0<|A|\leq \frac{|V|}{2}...
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Velcro surface: is it possible to have a surface which is everywhere infinitesimally a cone, but not a normed group?
Is there any example of this velcro-like space? Looking for a LOCALLY COMPACT COMPLETE metric space $(X,d)$ such that:
(A)-it has a metric tangent space $(T_{x}X, d^x)$ at any point $x \in X$,
(...
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3
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2
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301
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Optimizing the layout of Infinite Suburbia
Infinite Suburbia is a Euclidean plane, P. All residents live in open unit disks which, like caravans, can travel around but are stationary most of the time. When stationary, these disks lie in a ...
- 785
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4
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589
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Measures of the complexity of a metric
I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot. I am primarily interested in surfaces topologically
equivalent to ...
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14
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5
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Shortest-path Distances Determining the Metric?
The metric of a Riemannian manifold determines the shortest
distance between any two points.
I assume the reverse holds? That is, if you are given the
shortest distance d(x,y) between every pair of ...
- 151k
1
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1
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A question about dissecting spherical triangles
Do there exist spherical triangles which are not isoceles but are the union of a finite collection of
(two or more) congruent triangles with pairwise disjoint (and non-empty) interiors?
- 55.9k
7
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1
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Rolling a convex body: Geodesics vs. rolling curves
What are the curves of contact on a convex body $B$ rolling down an inclined plane?
Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic ...
- 22.3k
3
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1
answer
375
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Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
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0
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427
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The Gömböc and monostatic objects
This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I ...
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1
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335
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When is the neighbourhood of a set a ball?
In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
Proof: Let $S$ be contained in $B_r(y)$, $y \in \mathbb{...
- 55.9k
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1
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Fixed point theorems and equiangular lines
I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
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4
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1
answer
496
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Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 4,462
3
votes
0
answers
559
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Unprovability of the Steiner-Lehmus theorem
Conway postulated that the Steiner-Lehmus theorem is unprovable using direct methods of proof. Can this be proven directly, that the Steiner-Lehmus theorem cannot be proven directly over Euclidean ...
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1
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Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]
How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
- 7,105
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0
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335
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Irrationality of square root of 2 [closed]
It is possible to explain me the 18th proof of the irrationality of square root of 2 from the following site?
http://www.cut-the-knot.org/proofs/sq_root.shtml
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1
answer
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Finding integer points on an N-d convex hull
Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is ...
- 151k
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3
answers
1k
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Unit triangles with vertices on circles
Characterize all triples $c_1,c_2,c_3$ of circles in the plane such that
there are infinitely many unit regular triangles $a_1a_2a_3$ with $a_i\in c_i$ for $i=1,2,3$.
In particular, are there any ...
- 17.9k
8
votes
2
answers
621
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Generalization of Hamiltonian cycles to "Hamiltonian spheres"
One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be
called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$
...
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14
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7
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The Symmetry of a Soccer Ball
Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
An ...
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0
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579
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Cluster-preserving and distance-maximizing embedding into Hamming Space?
I have a set of data, each instance in the real $[0,1]^{d}$. However, it's actually all in a relatively small range around 0.5, clustered into classes in even smaller ranges. The actual origin of the ...
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3
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1
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152
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Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 44.4k
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0
answers
176
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Spaces with the thin tetrahedra property
I read a comment about the $\delta$-thin tetrahedra property of a space.
It basically means, that if you choose any four points in this space, connect them by geodesics, and fill each triangle with a ...
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2
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2
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215
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Is this a correct interpretation of support in coarse geometry?
Let $X = \mathbb{R}^n$, and consider a nondegenerate representation $\rho: C_0(X) \to B(H)$ where $B(H)$ is the algebra of bounded operators on a separable Hilbert space. The support of a vector $v \...
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1
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Hyperbolic structure on surfaces with boundary
I have following two questions
1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). ...
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14
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1
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587
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Pushing convex bodies together
Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies ...
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2
answers
176
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Polygon Chain - Conversion to non-crossing while preserving shape?
I have polygon chains similar to the following...
http://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Self_crossed_polygonal_chain.svg/220px-Self_crossed_polygonal_chain.svg.png
...given the ...
- 18.6k
4
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2
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789
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Is there any documented study of geometry in contemporary primates ? [closed]
There are many studies of language learning abilities of primates (mostly chimpanzee, bonobo) and studies of tool use, knowledge transmission and number sense.
Are there studies or documented cases ...
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1
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556
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A variation on "Hearing the shape of a drum" for polytopes.
Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...
CommunityBot
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Dissecting trapezoids into triangles of equal area
[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
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1
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1
answer
419
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Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 13.6k
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4
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How to pick a random direction in n-dimensional space
I want to pick a random direction in n-dimensional space. How can I do this?
The reason I want to do this is to pick a neighbor for hill climbing optimization.
- 1,944
8
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1
answer
398
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Möbius-invariant triangle center?
Given any two points x and y on a circle O, one can form four different lenses (regions between two circles, one of which is O) that have corners at x and y and make angles of 2π/3 at their corners. ...
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5
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1
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Do continuous maps give continuity in the 'topology' of Hausdorff distance?
I was reading this question:
limiting behaviour of converging loops on a torus
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
CommunityBot
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4
votes
2
answers
271
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Centralizing four red vectors in six green sectors
Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid star of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
\...
- 18.6k
1
vote
2
answers
1k
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Calculating the surface area distribution of two-dimensional projections for a polytope
My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex ...
CommunityBot
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6
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3
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982
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Boolean network as a gauge field
Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two ...
- 2,179
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2
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1k
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Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
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What are trig classes like within a universe that's "noticeably" hyperbolic?
[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] ...
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