Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,263
questions
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Metric space whose bounded subsets are totally bounded
Is there a name for a metric space in which any bounded subset is totally bounded, or equivalently, in which any bounded sequence contains a Cauchy subsequence?
I have seen the name Bolzano-Weierstraß ...
5
votes
1
answer
215
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Proof of Lemma 37.5 in Pak's Lectures on Discrete and Polyhedral Geometry
I am staring at the proof of Lemma 37.5 in Lectures on Discrete and Polyhedral Geometry, see page 331.
I cannot understand why the required triangulation exists.
In the first paragraph it says "...
7
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2
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167
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Bisector of two points in a Riemannian manifold has measure $0$
Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...
10
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0
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196
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Do cut-length-minimizing equidissections exist?
Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
3
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1
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Constrained morphing of polygons
This post continues 'Constrained morphing' of planar convex regions
If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one ...
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0
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156
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Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
2
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1
answer
82
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'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
3
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1
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Dividing a spherical cap into $n$ equal wedges
This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown.
Motivation: Optimal ways to cut an orange.
In this problem, we have a spherical ...
4
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0
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93
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Is there a 5-cell-600-cell honeycomb?
Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
1
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0
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Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
4
votes
2
answers
482
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On diffeomorphisms that preserve the metric
Suppose $\Omega\subset \mathbb R^2$ is a bounded domain with smooth boundary and suppose that
$$ F: \Omega \to \Omega,$$
is a diffeomorphism that fixes $\partial \Omega$ (i.e $F|_{\partial \Omega}$ is ...
1
vote
1
answer
263
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Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
2
votes
1
answer
103
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Lipschitz maps with Hölder inverse preserve the doubling property
Let $K$ be a compact doubling metric space, $X$ be a metric space and $f:K\rightarrow X$ be Lipschitz with $\alpha$-Hölder inverse, where $0<\alpha<1$. Does $f(K)$ need to be doubling?
11
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1
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466
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A topological tree is weakly contractible
Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
14
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0
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261
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Regular $n$-gon with diagonals: bounds on area of largest cell?
Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
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2
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153
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Condition to be concyclic [closed]
What condition would you impose upon $n$ points on a plane of which no three points are collinear so that they are
concyclic if the distances of each point from the all remaining points are known? (...
3
votes
1
answer
64
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Comparing the areas of polygons via equidecomposability in the hyperbolic plane
It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.
Question. Is an ...
2
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0
answers
62
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On local-to-global theorem of $\mathrm{CD}^*(K,N)$ spaces
In the 2010 JFA paper "Localization and tensorization properties of the curvature-dimension condition for metric measure spaces" (arXiv link, DOI link), the authors used Theorem 5.1 to prove ...
1
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1
answer
193
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Local Lipschitz constant of exponential map for Hadamard manifolds
Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local ...
5
votes
1
answer
336
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How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
14
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1
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272
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How many distances are required to calculate all distances among $n$ points in the Euclidean plane?
I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
5
votes
1
answer
201
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Covering unit-radius balls with unit-diameter objects
Let $d$ be a norm-based metric in $\mathbb{R}^2$. We are given a $d$-ball with radius 1, and we would like to cover it with objects with diameter 1. How many objects are needed?
In the $\ell_1$ metric,...
11
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3
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What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
9
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1
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368
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A name for a mathematical structure of geometric type
I am looking for (maybe existing) name for a mathematical structure $(X,\leqslant)$ consisting of a set $X$ and a transitive relation ${\leqslant}\subseteq X^2\times X^2$ such that $xx\leqslant yz\...
2
votes
0
answers
199
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Geometric inequality related with convexity of the boundary
I'm new to Mathoverflow, so hopefully my question is well-posed.
My problem states as follows:
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...
14
votes
2
answers
666
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How to characterize the regularity of a polygon?
In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
5
votes
2
answers
759
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Checking that the image of a curve is not contained in a hyperplane
Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear ...
2
votes
1
answer
144
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Looking for the 3D-analog/extension of a 2D theorem
The excentral triangle of a reference triangle $ABC$ is the triangle with vertices corresponding to the excenters of $ABC$. Denote with $D$, $E$, $F$ the $A$−, $B$−, $C$− excenters, respectively. ...
2
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0
answers
70
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Is parquetability decidable?
Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is parquettable by $T$ if there is a partition $\frak P$ of $\mathbb{Z}...
1
vote
1
answer
150
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Right-continuity of covering number
Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
1
vote
1
answer
172
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What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
3
votes
0
answers
112
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Can a laser hit all the mirrors out of order?
For this question, a "cycle" is a sequence of distinct points
$X = (x_1,x_2,\cdots,x_k)\in\mathbb{R}^3$ which defines a piecewise linear path starting at $x_1$ and visiting the points in ...
4
votes
0
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130
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A Lipschitzian's condition for the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...
4
votes
0
answers
113
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Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$
The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...
4
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2
answers
244
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Discrete isoperimetric problems
It is well-known that among all planar curves, the circle — invariant under $O(2)$ — has the best isoperimetric ratio. Similarly, among all $n$-gons, the regular $n$-gon — invariant under the dihedral ...
3
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0
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168
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Short selection in the space of subsets
Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric.
Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a ...
2
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0
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81
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Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
1
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0
answers
142
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what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
5
votes
1
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204
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Which pyramids fill space?
Let us define a pyramid as a convex polyhedron with one quadrilateral face and four triangular faces.
Question: How many pyramids (or families of pyramids) are known that can fill 3D space without ...
1
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0
answers
93
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About semiconvex and semiconcave functions in geodesically complete CAT(0) spaces
I posted this question a while ago but had no luck. So I'm trying again. Let $(X,d)$ be a proper, geodesically complete $CAT(0)$ space and let $\Sigma_x X$ be the space of directions at a point $x\in ...
17
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363
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Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
1
vote
0
answers
104
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John and Lowner ellipsoid
I am looking at a proof to show that Lowner ellipsoids are unique for centrally symmetric convex body $K$. I want to show basically that
$$
\DeclareMathOperator{\Vol}{Vol}\DeclareMathOperator{\Low}{...
1
vote
1
answer
91
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Intersection of conical neighbourhoods on a polyhedral space
Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...
5
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0
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204
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Does every smooth surface contain three congruent curves?
Let $M \subset \mathbb R^3$ be a smooth, connected, closed surface. We say a family of smooth curves on $M$ are congruent if each of them can be mapped to any other by a isometry of the ambient ...
1
vote
1
answer
133
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How to reduce the compact support to the case of small diameters in Tao's "A sharp bilinear restriction estimate for paraboloids"
I am reading Terence Tao's paper "A sharp bilinear restriction estimate for paraboloids"
to prove the bilinear restriction estimate on paraboloids. In Section 3, he assumes that $\text{diam}(...
2
votes
0
answers
72
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Hidden vector recovery problem?
A certain problem has shown up in my research recently that feels like it could be a preexisting problem in convex geometry, but I am not familiar with the area.
I'll describe the problem (and a ...
1
vote
0
answers
77
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Lattice packing
Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
3
votes
0
answers
187
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Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
3
votes
1
answer
112
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Distance of average of points to center of minimum enclosing ball
Let $v_1, ..., v_n$ be distinct points in $\{0,1\}^d$ with the same norm $\|v_i\|_2=k$ (i.e each $v_i$ has $k$ ones). Let $A=\frac{1}{n}\sum_{i=1}^n v_i$ be their average, and let $C$ be the center of ...
7
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0
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161
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Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...