Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,249
questions
2
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1
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368
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Minkowski sum, zonotopes, convex hull
For any set $P,Q$ in the Euclid space, define Minkowski sum '+' as follows: $P+Q=\{p+q|p\in P, q\in Q\}$. And define 'zonotope': a zonotope is the Minkowski sum of some (finite) segments (for example, ...
6
votes
0
answers
216
views
Is this function embeddable in Euclidean space?
Let $X = \{v_1,\ldots,v_n\}$ be a set of vectors non-zero vectors $v_i \ge 0$ and such that the vectors are pairwise linear independent. Define a function on this set $X$:
$$d(v,w) = 1-\frac{2 \...
4
votes
1
answer
271
views
Is the projection of a weakly Lipschitz domain still a Lipschitz domain?
We say, following this definition, that a domain $\Omega\subset \mathbb{R}^{n}$ is weakly Lipschitz if it can locally be flattened by a Lipschitz homeomomorphism $\phi$ (i.e., a Lipschitz continuous ...
3
votes
1
answer
156
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Is there a definition for "convexity" of spatial (non-planar) polygons? [closed]
I was thinking that there should exist a definition for "convexity" of spatial polygons.
A planar convex quadrilateral that has one vertex moved (perpendicularly) out of the plane should continue to ...
0
votes
1
answer
210
views
Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
9
votes
1
answer
686
views
Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
1
vote
0
answers
353
views
Geometric interpretation of metric [closed]
For metric in $\mathbb R^2$ (polar coordinates) Pythagorean triangle elements can be visualized in a differential form as seen at top yellow triangle. This happens for metric:
$$ ds^2= dr^2 +(r d \...
4
votes
1
answer
130
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Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?
This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
1
vote
1
answer
138
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Meaning of "quantitative result" [closed]
Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
1
vote
0
answers
84
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A Hölder version of the Johnson-Lindenstrauss Lemma on essentially bounded functions
Does there exist a Hölder (not necessarily linear) projection from $L^{\infty}(\mathbb{R}^d)$ to any finite-dimensional linear subspace? This is known when $L^{\infty}(\mathbb{R}^d)$ is replaced by a ...
5
votes
1
answer
188
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Is the barycenter of a convex plane curve Lipschitz with respect to the Hausdorff distance?
Crossposted from Math Stack Exchange
For a convex curve $C$, define its barycenter to be
$$b(C) = \frac{1}{\mathcal H^1(C)} \int\limits_C x d \mathcal H^1(x)$$
Is there a constant $L$ such that for $...
1
vote
1
answer
724
views
Known Lipschitz-free spaces
The Lipschitz-Free space (also known as Arens-Eells spaces) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of ...
5
votes
2
answers
487
views
Concrete description of lift in Arens-Eells space
Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of Nik Weaver shows that for every ...
5
votes
0
answers
270
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When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
24
votes
4
answers
2k
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A reinterpretation of the $abc$ - conjecture in terms of metric spaces?
I hope it is appropriate to ask this question here:
One formulation of the abc-conjecture is
$$ c < \text{rad}(abc)^2$$
where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
14
votes
1
answer
834
views
What are the applications of the Mazur-Ulam Theorem?
Every bijective isometry between normed spaces is affine. This well-known and beautiful statement, the Mazur-Ulam Theorem, was proved in 1932, but the proof has been simplified and polished in years, ...
-1
votes
2
answers
295
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A Erdős–Mordell Like inequality
Ono's inequality is true for acute triangle but false with general triangles. The inequality as follows is false with general triangls but I think it true with acute triangle (follows answer by Fedor ...
3
votes
1
answer
436
views
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 ...
3
votes
0
answers
102
views
"Hoelder conjugate" version of the Johnson-Lindenstrauss transform
A variation of the well-known Johnson-Lindenstrauss transform (JLT) asserts that for $x_1,\ldots,x_m\in\mathbb{R}^n$ there exists a linear transformation $A:\mathbb{R}^n\to\mathbb{R}^k$ with $k=\...
12
votes
2
answers
927
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Set of points with a unique closest point in a compact set
Let $K\subset\mathbb{R}^n$ be any compact set. Let $\operatorname{Unp}(K)$ be the set of points in
$$
\operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}.
$$
...
4
votes
0
answers
97
views
Is every locally compact connected homogeneous metric space a manifold cross a continuum?
Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$...
5
votes
1
answer
785
views
To find the Largest Regular n-gon contained in a given convex region
Given a general convex region C, to find the largest regular polygon that is contained in it (shared boundaries allowed). Basically, one needs to find that particular value of n for which a regular n-...
5
votes
0
answers
137
views
Mordell's theorem on rational quadrilaterals
Mordell proved that for any epsilon and any quadrilateral in the Euclidean plane there is an epsilon-close quadrilateral whose sides and diagonal are rational. Does this break down for five points in ...
1
vote
1
answer
143
views
A Uniform Metric Selection Theorem
Let $X$ and $Y$ be bounded complete separable metric spaces. Let $C = 2^\omega$ be Cantor space with its standard metric. All product spaces are taken to have the max metric.
Let $F, G \subseteq X\...
3
votes
1
answer
149
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
7
votes
2
answers
529
views
Do Minkowski sums have anything like calculus?
Is there anything resembling differential calculus over the space of (nicely behaved) regions in $\mathbb{R}^d$, where addition is interpreted in terms of Minkowski sums? For example, it is known ...
9
votes
1
answer
219
views
Isometries of convex hypersurfaces
The well known Pogorelov’s rigidity theorem says that if two convex closed surfaces in Euclidean 3-space are isometric to each other being equipped with their intrinsic metrics, then they are ...
2
votes
1
answer
467
views
Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
1
vote
1
answer
396
views
Prospects for deep learning of non-lattice sphere packings
I have been looking for litterature on results obtained by deep neural networks to find dense (and quite possibly non-lattice, perhaps even non-periodic) sphere packings, but I have not been too ...
6
votes
1
answer
631
views
Napkin Folding Problem / Rumpled Ruble Problem
I am an outsider to this group. I'm a journalist and am working on a piece about theoretical math/geometry. Simply put, when a napkin is folder in such a way to increase its perimeter is that strictly ...
2
votes
1
answer
363
views
Gromov-Hausdorff distance between weighted tree graphs
I would like to measure the similarity between a pair of weighted tree graphs. According to this post, this can be done by regarding the trees as metric spaces and then applying the Gromov-Hausdorff ...
1
vote
1
answer
100
views
Lipschitz vs. bi-Lipschitz parametrizations for subsets of Euclidean space [closed]
Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is ...
1
vote
0
answers
141
views
A topological property of curves on the plane $\mathbb{R}^2$
Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous injective map.
Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\...
12
votes
2
answers
416
views
How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
3
votes
1
answer
176
views
Dimension of Alexandrov space which is homeomorphic to a manifold
Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$.
It is true that the ...
7
votes
1
answer
482
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 ...
-1
votes
1
answer
117
views
Horospherical distance in CAT($-1$) space
In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d_{\...
4
votes
0
answers
77
views
Proximal isometries in CAT($-1$) metric space
Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
5
votes
1
answer
538
views
Fast Bourgain embedding (or similar embeddings)?
Currently I am working on applications of Bourgain Embedding (or similar embeddings of finite metric spaces to $l_2$) to automatic feature engineering for machine learning/data science ( http://www....
2
votes
2
answers
262
views
Existence of a Hölder-free space
The Lipschitz-free or Arens-Eells space over a pointed separable metric space $(X,0,d)$ is a well-studied object. My question is, is an analogos Hölder-free space; for a fixed Hölder constant $\alpha&...
7
votes
1
answer
694
views
To minimize the Hausdorff distance between convex polygonal regions
Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given two convex polygonal regions P1 and P2 on the ...
0
votes
1
answer
167
views
Does every compact doubling metric space have a canonical measure?
My question is this one, with the additional condition that the metric space be doubling. In the aforementioned question, the limiting measure depends on the sequence $\epsilon_n$ and hence is not ...
10
votes
2
answers
363
views
How many small dots can be drawn in a region such that no three are "collinear"?
When people draw dots on paper, they are actually not points, but small regions filled with ink. Suppose that each dot has disc-shape with fixed radius $r\ll 1$ and must be drawn inside (1) a square ...
3
votes
0
answers
142
views
Upper bound on the geodesic distance in a Lipschitz domain
I was wondering if the following result is true. If yes, could you please suggest a reference. The result seems to have been used at several papers without quoting any reference. Is the proof ...
0
votes
1
answer
109
views
Neighborhood of $(n,\delta)$-strained point in Alexandrov space homeomorphic to $\mathbb{R}^n$, how big is $\delta$?
Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geqslant k$. A point $p\in M$ is said to be an $(n,\delta)$ strained point if there are $n$ pairs of points $a_i, b_i$ such that
$$
\...
9
votes
0
answers
203
views
A geometric characterization of quasicircles
I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...
9
votes
2
answers
542
views
Every riemannian length structure on $\mathbb{R}^n$ is induced by a continuous function $f:\mathbb{R}^n\to \mathbb{E}^n$, to the euclidean space
This question is a cross post from Math.SE. Unfortunately the migration of the question is not possible after two months of posting.
I have been reading about length spaces in the (great) book Metric ...
4
votes
1
answer
316
views
Can planar set contain even many vertices of every unit equilateral triangle?
Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
3
votes
0
answers
180
views
Pointed version of Perelman stability theorem
I am wondering if there is a version of the Perelman stability theorem which says approximately the following:
Let $\{(X_i,p_i)\}$ be a sequence of pointed $n$-dimensional complete Alexandrov spaces ...
3
votes
1
answer
312
views
How can the same polytope have three different volumes? [closed]
I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the permutohedron, formed by the convex hull of the n! points ...