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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Lemoine-Lozada circles
I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
- 445
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176
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Subsets of a ball/sphere with the largest sum of distances
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B_d$ and $S_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B_d$ is ...
- 128k
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Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 42k
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Rademacher type theorem for Alexandrov spaces
The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere.
I am wondering if this result can be generalized as follows. ...
- 21.8k
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Harmonic maps are light
Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I ...
- 45k
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839
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Can the projection onto a compact set always be taken to be measurable?
This may be a very basic question.
Let $X$ be a complete metric space and let $T$ be a compact subset of $X$. Say that a function $\pi: X \to T$ is a projection if
$$
d(x, \pi(x)) = d(x, T) \quad \...
- 410
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Separating pairs of points in R^n
Let $A$ be a set of $2k$ points in $\mathbb{R}^n$ such that no open set in $\mathbb{R}^n$ of diameter $2$ contains more than $k$ of these points. What is the largest possible distance $r_n>0$ one ...
- 2,288
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Fattening of totally convex sets
Suppose $(M, g)$ is an open complete nonnegatively curved Riemannian manifold with $d$ its distance.
A totally convex set $C\subset M$ has the property that for any two point $x, y \in C$ any ...
- 1,309
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3
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482
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Herringbone partitions of regions and surfaces
Let $R \subset \mathbb{R}^2$ be a region of the plane bounded
by a Jordan curve. The boundary $\partial R$ could be a polygon,
or a smooth curve—there are variations depending upon boundary ...
- 151k
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What is the origin of unit vector notation? (i,j,k)
What is the origin of this notation? Who coined them and for what purpose?
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Light rays bouncing around inside a sphere in d-dimensions
Suppose $S=\mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center ...
- 151k
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Minimum spanning tree of a weighted graph
I have a connected graph $G=(V,E)$ in $n$ vertices. The edge weights are non-negative and form a metric space, thus for vertices $u,v,w \in V$ , such that $(u,v), (v,w), (w,u)\in E$ we have $r(u,w) \...
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Generalizing cosine rule to symmetric spaces
The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
user2529
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Minimum-area bounding quadrilateral algorithm
There are a few algorithms around for finding the minimal bounding rectangle (OBB) containing a given (convex) polygon.
Does anybody know about an algorithm for finding a minimal-area bounding ...
- 163
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3
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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 ...
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Mutual metric projection
Given a subset $S\subseteq \mathbb{R}^n$, the metric projection associated with $S$ is a function that maps each point $x\in \mathbb{R}^n$ to the set of nearest elements in $S$, that is $p_S(x) = \arg ...
- 3,559
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Concentration of volume towards the boundary
Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all ...
- 93
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Sphere packing processes during biological development
Within the context of mathematical biology, a sphere packing problem occurred to me. I must note that unlike the typical sphere packing problems, the variant I consider involves minimising the average ...
- 3,871
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Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space
Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure.
I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
- 5,405
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761
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Embedding metric spaces into Hilbert ones
Does every compact metric space continuously embed into a Hilbert space (possibly with large distortion)?
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412
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What is the geometric meaning of one Riemannian metric bigger than the other one on a smooth manifold?
Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics ...
- 1,799
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Discrete approximations of Riemannian manifolds
MSE crosspost
It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\...
- 4,600
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Bichromatic pencils
A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...
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A class of tilings with amazing visual qualities
For more examples please see my related question on MSE:
Interesting tiling with a lot of symmetrical shapes
This is achieved by rotation of square grid over itself by atan(3/4).
Resulting ...
- 161
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880
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Relation of some Euclidean geometry theorems and more conjecture generalizations
In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem, the Ptolemy's theorem and the Feuerbach-Luchterhand. Since ...
- 1,044
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Gauss-Bonnet formula for 2-dimensional Alexandrov spaces
EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.
Questions 1. Can one define a measure $K$ on $S$ (thought ...
- 21.8k
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555
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Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?
Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true ...
- 6,034
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The reflex-free hull: Construction?
This is a followup to Bill Thurston's question
about different notions of hulls.
Here I want to raise a question about the
reflex-free hull, which is, intuitively, the smallest
enclosing shape to an ...
- 151k
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715
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Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
- 1,318
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
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435
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On the aperiodic monotile
One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
- 4,898
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Can a dodecahedron be deformed into a great stellated dodecahedron?
Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
- 2,773
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Like Bradley’s conjecture (Four incenters lie on a circle) [closed]
Please don't close this question. Because there is simple configuration with 57 vote up, and don't close. Why you vote up that question and You vote to close this question?
A problem I posed at here ...
- 1,776
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A polytope with congruent facets and an insphere that is not facet-transitive?
Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
All facets are congruent,
it has an ...
- 13.6k
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211
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An arrangement of entangled squares
Is there an arrangement of finitely many axes-parallel squares in the plane, of $k$ different colors, such that:
The squares of each color are pairwise-disjoint;
Each square overlaps at least $4$ ...
- 3,559
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2
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Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
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691
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Triangle angle bisectors, trisectors, quadrisectors, …
With the triangle
angle bisector theorem
and
Morley's trisector theorem
as background ,
are there any pretty theorems known for triangle $n$-sectors,
$n > 3$?
For example, angle quadrisectors?
The ...
- 151k
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178
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Point distributions in unit square which minimize E[1 / distance]
Choose $n$ points $p_1,\ldots,p_n$ in the unit square $[0,1]^2\subset\mathbb{R}^2$
such that $D:=\mathop{\sum}\limits_{1\le i<j\le n}\frac{1}{dist(p_i,p_j)}$ is minimized, where $dist(p_i,p_j)$ is ...
- 71
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Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
- 151k
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$L^{p}$ isoperimetric inequalities on the Hamming cube
Let $A \subset \{-1,1\}^{n}$ be a subset of the Hamming cube with cardinality $|A|=2^{n-1}$. Define $w_{A} : \{-1,1\}^{n} \to \mathbb{N}\cup \{0\}$ so that $w_{A}(x)$ to be number of boundary edges ...
- 3,946
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Problem on triangles
Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...
user101163
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Union of almost Hamming balls
This question is connected with my previous question: Union of Hamming balls
Let $V \subseteq \{0,1\}^n$, $\log|V| = k < 0.9n$.
Harper's theorem states that the set $V_r:= \bigcup_{x \in V} V_r(...
- 2,576
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247
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Geometry of convex subsets in Alexandrov space/ Riemannian manifold
Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...
- 21.8k
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Self-avoiding/reflecting geodesics on a convex surface
Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...
- 151k
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347
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Measurement of "symmetry" of a convex body
I often hear that the regular simplex is "the least" symmetric convex body, and I've heard that there are some measures of symmetry of a body, that the simplex minimizes.
Could you please explain or ...
- 1,893
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268
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Counting valid coordinates
We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in \...
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Matching on sphere to create cycle with chords
Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like to ...
- 151k
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1
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249
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What is the doubling dimension of convex functions?
I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^...
- 173
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147
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On multi-dimensional real trees
A real tree is a metric space $(M,d)$ satisfying the following two conditions:
(1) for every $x,y\in M$, there is an unique isometry $\phi$ from the closed interval $[0,d(x,y)]$ onto $M$ such that $\...
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