Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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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, ...
Yachy's user avatar
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6 votes
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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 \...
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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 ...
Gil Sanders's user avatar
3 votes
1 answer
156 views

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 ...
elwyn's user avatar
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0 votes
1 answer
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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 ...
ABIM's user avatar
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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-...
Chris Z's user avatar
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1 vote
0 answers
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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 \...
Narasimham's user avatar
4 votes
1 answer
130 views

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 ...
James Hanson's user avatar
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1 vote
1 answer
138 views

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.
ABIM's user avatar
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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 ...
ABIM's user avatar
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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 $...
Justthisguy's user avatar
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 ...
ABIM's user avatar
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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 ...
ABIM's user avatar
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5 votes
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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 ...
Paul Cusson's user avatar
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24 votes
4 answers
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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 ...
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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, ...
Pietro Majer's user avatar
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-1 votes
2 answers
295 views

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 ...
Đào Thanh Oai's user avatar
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 ...
Nandakumar R's user avatar
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3 votes
0 answers
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"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=\...
user134977's user avatar
12 votes
2 answers
927 views

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 $$ \operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}. $$ ...
Piotr Hajlasz's user avatar
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 $f(x_0)=x_1$...
James Hanson's user avatar
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5 votes
1 answer
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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-...
Nandakumar R's user avatar
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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 ...
Peter Kropholler's user avatar
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\...
James Hanson's user avatar
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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 ...
M. Winter's user avatar
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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 ...
James Ingram's user avatar
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 ...
asv's user avatar
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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 $\...
Nandakumar R's user avatar
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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 ...
Archie's user avatar
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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 ...
user145712's user avatar
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 ...
edelburg's user avatar
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 ...
mdr's user avatar
  • 527
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\...
asv's user avatar
  • 21.1k
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 ...
Nikita Kalinin's user avatar
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 ...
asv's user avatar
  • 21.1k
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 ...
Joseph O'Rourke's user avatar
-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_{\...
user470881's user avatar
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 ...
user470881's user avatar
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....
user avatar
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&...
ABIM's user avatar
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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 ...
Nandakumar R's user avatar
  • 5,401
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 ...
Aryeh Kontorovich's user avatar
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 ...
Haoran Chen's user avatar
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 ...
Tatin's user avatar
  • 895
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 $$ \...
mathmetricgeometry's user avatar
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 ...
sharpe's user avatar
  • 701
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 ...
Dante Grevino's user avatar
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 ...
domotorp's user avatar
  • 18.3k
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 ...
asv's user avatar
  • 21.1k
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 ...
RMurphy's user avatar
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