Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,406 questions
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Maximal set on hypersphere that does not contain pairs of orthogonal vectors
Let R be a region on a hypersphere. Each point A of the hypersphere
is associated with a vector pointing to A and with origin at
the centre of the hypersphere. So let me identify each point with a
...
- 1,207
8
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0
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358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
8
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0
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588
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Hausdorff measure question
Say we have some compact metrisable topological space $X$ with a measure $\mu$ defined on the Borel sets of $X$. Then is there some way to determine whether $\mu$ is the Hausdorff measure associated ...
- 1,665
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6
answers
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How to partition R^3 into pairwise non-parallel lines?
Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...
- 1,110
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4
answers
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Do cotangent bundles have "bounded geometry"?
I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and $\mathbb{R}^n$...
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Bijective function on a dense set
Suppose X is a complete metric space, and $f:X↦X$ a continuous surjective function. Let D be a dense set. Suppose $f:D↦D$ is injective and $f^{-1}(D)=D$.
Is $f$ injective ?
Is there a family of ...
- 307
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votes
2
answers
805
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Distance among integer set
Given an integer set, if the distances between integers in the set are still in the set, what mathematical term should be used to describe that nature? Or what nature does the set have?
For example, $...
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2
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711
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Does list of distances define points uniquely?
There are N points on a plane. Is it feasible to reproduce their relative location
having only the list of distances. Assuming that translation, rotation and mirror are allowed
in the result. The ...
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3
answers
510
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Do sufficiently regular distances on manifolds come from riemannian metrics?
Hi to all!
Let $M$ be a compact smooth manifold without boundary and let $$d:M\times M\rightarrow [0,+\infty)$$ a distance on $M$ compatible with its topology. Suppose there exist $\varepsilon\in (0,+...
- 1,727
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votes
2
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539
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Tangled Knot Function
I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$
that has these properties:
(1) When iterated $n$ times starting from some $p$,
connecting the points in order
with segments and closing ...
- 151k
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votes
3
answers
420
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Sectional curvature of leaves of foliation
Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ ...
- 229
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votes
3
answers
866
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Not quite regular polyhedra
Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...
- 1,314
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votes
4
answers
723
views
Surfaces that can be rolled by a ball
Let $S$ be a smooth solid body in $\mathbb{R}^3$,
and $B$ a ball of radius $r$.
Say that $B$ is in contact with $S$ if
(1) they share a point $x$
that is on the surface of each,
$x \in \partial S$ ...
- 151k
7
votes
3
answers
792
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Shadow boundary on convex body in $\mathbb{R}^3$
Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$,
with strictly positive Gaussian curvature at every point of $S$.
Fix a direction $z$ in a Cartesian coordinate ...
- 151k
7
votes
2
answers
529
views
What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
- 27.7k
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2
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549
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Kissing Number of Spheres in Non-Euclidean Geometry
There has been much work done on the kissing number problem (of determining the greatest number of congruent spheres which can touch a single sphere in a packing) in Euclidean space for dimensions $1$ ...
- 1,441
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votes
1
answer
686
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Regular simplex in projective space
Is there a reference or a very short argument proving the following statement?
Let $C$ be a set consisting of $r$ points in the real projective space $\mathbb RP ^k$
with its usual round metric. ...
7
votes
2
answers
279
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Kissing number and overlapping number
Let $S$ be a certain family of geometric objects (e.g, the family of unit squares).
The kissing number of $S$ is the maximum number of nonoverlapping elements of $S$ that can touch one element of $S$...
- 3,559
7
votes
2
answers
504
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Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?
If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...
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7
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2
answers
299
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subsets of products of trees
A subset of a geodesic metric space is called convex if for every two points in the subset one of the geodesics connecting these points lies in the subset. Is it true that every convex subset of a ...
user6976
7
votes
3
answers
401
views
Maximizing the area of a region involving triangles
I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
- 26.9k
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2
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244
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approximate two different real numbers to order $\frac{1}{z^{3/2}}$
I took this result from Minkowski's book on Geometry of numbers:
Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and $\...
- 22.8k
7
votes
2
answers
1k
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A characterization of Hilbert spaces?
My question was prompted by an earlier MO by @Daniel:
Duality map in strictly convex Banach spaces
I will even use his symbol $\phi$ below.
Let $B$ be an ...
- 7,500
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votes
4
answers
3k
views
completeness axiom for the real numbers
Do any treatises on real analysis take the following as the basic completeness axiom for the reals?
"Let $A$ and $B$ be set of real numbers such that
(a) every real number is either in $A$ or in $B$;
...
- 19.7k
7
votes
1
answer
483
views
Furthest distance half the diameter?
Let $S$ be the surface of a convex body, polyhedral or smooth,
embedded in $\mathbb{R}^3$.
For a point $x \in S$, let $F(x)$ be the set of furthest points
from $x$, measured by shortest paths on the ...
- 151k
7
votes
3
answers
678
views
How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
- 8,086
7
votes
2
answers
1k
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Is a given point in the interior of the convex hull of a given finite collection of points?
Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
- 96.4k
7
votes
3
answers
511
views
Proto-Euclidean algorithm
Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining
$$q_i = \left\lfloor \frac{r_i}{r_{...
- 9,720
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 5,063
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votes
2
answers
1k
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G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
- 579
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4
answers
3k
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Existence of Fermi coordinates on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
- 8,512
7
votes
2
answers
1k
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Example of non-closed convex hull in a CAT(0) space
this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
- 4,462
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votes
2
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355
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Convex subcomplexes of CAT(0) cubical complexes
Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...
- 8,493
7
votes
1
answer
962
views
Which surfaces have only a finite number of connecting geodesics?
Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$,
under which conditions is it true that, for every pair of points
$a,b \in S$, there are an infinite number of ...
- 151k
7
votes
1
answer
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The relation between Hausdorff dimension of an $n$-manifold and $n$
It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance)...
- 285
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votes
1
answer
293
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Can a tangle of arcs interlock in plane?
This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here:
Can a (finite) collection of disjoint circle arcs in $\...
- 4,312
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votes
5
answers
5k
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Distance of a barycentric coordinate from a triangle vertex
I have a triangle $ABC$ with side lengths $a,b,c$ (edges $BC, CA, AB$ respectively).
I have a point $p$ with barycentric coordinates $u:v:w$.
These are normalised: $u+v+w=1$.
$1:0:0$ corresponds to ...
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votes
1
answer
865
views
Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
- 109k
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5
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How to compute the average distance till intersection within a triangle in $\mathbb{R}^2$?
You are given 3 points in $\mathbb{R}^2$; $A$, $B$, $C$ forming a triangle with area > 0. You pick an arbitrary point inside $ABC$ and an arbitrary direction. After some distance $d$, you will ...
- 171
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votes
2
answers
546
views
Doubling dimension vs other metric dimensions
For separable metric spaces, three fundamental notions of dimension
are equivalent:
$$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$
Where does the doubling dimension
fit into the picture?
- 6,541
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1
answer
1k
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A problem on infinite dimensional metric space
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$...
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votes
3
answers
2k
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Are properties of geodesics on a cylinder unique to cylinders?
The geodesics on a cylinder (a cylinder infinite in both directions) are either
(1) simple (non-self-intersecting) closed geodesics, or
(2) simple infinitely long geodesics (infinite in both ...
- 151k
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votes
2
answers
595
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cyclic polygons & trigonometry
I posted this question to stackexchange, where it's generated some comments but no progress toward answering it. I'm going to say somewhat more here than I did there.
At one vertex of a pentagon ...
7
votes
2
answers
366
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Simplicial and cubical decompositions of low valence
Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...
- 28.9k
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1
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943
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Relation between Ricci curvature and sectional curvature for 3-manifolds
Let $(M^n,g)$ be a smooth Riemannian manifold. It is well known that if $sec(M)\geq \kappa$ then $Ric(M)\geq (n-1)\kappa$.
If I understand correctly in dimensions $n\geq 4$ a lower bound on $Ric(M)$ ...
- 21.8k
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votes
3
answers
474
views
Polygonal paths and polygons with prescribed set of vertices
Let $A$ be a finite set of points in the plane. How can we determine if there is a simple open polygonal path (i.e. without intersections), whose vertices are exactly $A$, with no straight angles ...
- 329
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votes
2
answers
434
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Convexity in co-ordinate charts of geodesic balls
Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...
- 3,212
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votes
1
answer
679
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A problem of four conics
I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
- 1,776
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votes
1
answer
145
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Monotonicity of canonical ellipsoids
Let $\mathcal{C}$ be the set of compact convex centrally symmetric sets in $\mathbb{R}^d$, and let $\mathcal{E} \subset \mathcal{C}$ be the set of ellipsoids centered at the origin.
I'm looking for a ...
- 2,479