Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
667
questions
9
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maximum sum of angles between $n$ lines
Take $n$ lines in $\mathbb{R}^d$ (not necessary different, and all passing through the origin, though this is not important). What is maximal possible sum of angles between them for given $n$ and $d$? ...
- 103k
9
votes
2
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569
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Unknown work of Nöbeling on topological/Hausdorff dimension
Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to ...
- 27.3k
9
votes
2
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907
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Dense sphere packings which are not lattice packings
This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
- 3,975
9
votes
2
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701
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Generalization of Pascal's theorem to higher dimensions
Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
- 4,454
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1
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720
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Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
- 1,599
8
votes
1
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860
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Maximal tetrahedra inscribed in ellipsoid
Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...
- 149k
8
votes
1
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403
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Areas of Triangles in (Non-Riemannian) Metric spaces?
I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this ...
- 15.4k
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1k
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Differentiability of distance to a closed convex set [closed]
Let $( \mathbb{R}^d, \| \mathbf{x}\|_2 )$ be a Euclidean Space. For any nonempty closed convex set $A\subseteq \mathbb{R}^d$, we define
\begin{align}
d(\mathbf{x}, A) = \inf \{ \| \mathbf{x} - \mathbf{...
- 1,127
8
votes
3
answers
536
views
Set of vectors separated by at least a specified angle
Suppose $\theta$ and $d$ are given.
How big can a set of $d$-dimensional vectors be such that no pair of them are at angle less than theta?
I particularly want an upper bound; that is, an $n=n(\...
8
votes
2
answers
1k
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Maximal number of connected components of complement to an affine plane real algebraic curve
Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
- 403
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2
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467
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Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension
In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the ...
- 339
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votes
1
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174
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$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary
Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. ...
- 1,677
7
votes
5
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2k
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Surface area of superellipsoid (dice)
I'm a physical chemist and I am involved in “colloidal dice”. These are small, cube-like particles with a really nice, regular shape. These particles are not really cubic, but more rounded, much like ...
- 73
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3
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651
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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 ...
- 8,046
7
votes
2
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560
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Cutting a spherical surface into mutually non-congruent pieces of equal area
Question: For what values of integer $n$ can the surface of a sphere be partitioned into $n$ convex and mutually non-congruent pieces of same area? (convexity could be viewed as geodesic convexity). ...
- 5,493
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0
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404
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Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]
Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...
- 477
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2
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377
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A question about a question about 3-dimensional convex bodies
For each positive integer n let E(n) denote n-dimensional Euclidean space and let the term "n-dimensional convex body" mean a compact convex subset of E(n) whose interior (with respect to E(n)) is non-...
- 6,105
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2
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1k
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A conjecture generalization of Karamata inequality
Fist I observe function $f(x)=x^2$ in the figure as following
I found that when $x_1 \ge y_1$ and $x_2 \le y_2$ $\Rightarrow$ $AB \ge CD$
$\Rightarrow$ $$\frac{f(x_1)+f(x_2)}{2}-f(\frac{x_1+x_2}...
- 1,034
7
votes
1
answer
488
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map from 6-vertex model to domino tiling
I am trying to find a correspondence between 6-vertex model and an Aztec Diamond tiling. Here are the building blocks of the 8-vertex model:
There seems to be more than one correspondence. I found ...
- 22.6k
7
votes
1
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862
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Lebesgue differentiation theorem holds on locally doubling space?
It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...
- 471
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5k
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Shrink polygon to a specific area by offsetting
I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...
- 171
7
votes
2
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595
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What is the name for a set endowed with a Lipschitz structure?
I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
- 40.9k
6
votes
2
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523
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Volume ratio of general $\ell_p$ balls and surfaces
This question is a generalization of the question Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces
For any $p\in[1,\infty]$ define $\|x\|_p := (|x_1|^p+\cdots+|x_d|^p)^{1/p}$ for $p\in[1,\infty)$ ...
- 373
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votes
1
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3k
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Covering number of Lipschitz functions
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
That the $\infty$-...
- 2,146
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votes
0
answers
112
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How many equilaterals have vertices intersections of angle trisectors of a triangle?
The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral.
In the paper Trisectors like Bisectors with Equilaterals ...
6
votes
0
answers
176
views
Optimal planar net for catching convex shapes
Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...
- 149k
6
votes
4
answers
2k
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Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
- 11.9k
6
votes
1
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426
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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 ...
- 18.4k
6
votes
1
answer
2k
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Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?
Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
- 163
6
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1
answer
790
views
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 and the Feuerbach-Luchterhand. Since that I posed two ...
- 1,034
6
votes
1
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756
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Using mirrors to make a non-convex polygon visible from a fixed interior point
Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
- 17.4k
6
votes
0
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146
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Does every Tarski plane embed into a 3-dimensional Tarski space?
By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
- 40.9k
6
votes
1
answer
353
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Symmetric matrices and Hilbert's fourth problem
From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
\...
- 13.2k
6
votes
1
answer
447
views
Cutting the unit square into pieces with rational length sides
The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational.
To cut a unit square into n (a finite ...
- 5,493
5
votes
1
answer
418
views
Golden ratio as a property of conic section (is it known?)
I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...
- 1,086
5
votes
1
answer
153
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On folding a polygonal sheet
Consider a polygonal sheet $P$ of area $A$ with $N$ vertices (it material is not stretchable or tearable). Let $n$ be a positive integer >=2.
Question: Let $P$ lie on a flat plane. We need to fold ...
- 5,493
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votes
1
answer
183
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Which Banach spaces are absolute Lipschitz extensors for compacta?
A metric space $X$ is defined to be an absolute Lipschitz extensor for compacta if each Lipschitz map $f:K\to K$ defined on a compact subset $K\subset X$ extends to a Lipschitz map $\bar f: X\to X$.
...
- 40.9k
5
votes
1
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174
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Maximal regions with given diameter
Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and
$\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$.
Is it possible to describe all maximal regions?
The only ...
- 88.8k
5
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0
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329
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$N$-$th$ closed chain of six circles
Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
- 1,086
4
votes
0
answers
113
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Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,345
4
votes
1
answer
382
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Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?
Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
4
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0
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143
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Is the orthocenter "(roughly) equationally finitely-based"?
Let $T$ be the "almost everywhere" equational theory of the orthocenter function, "tweaked appropriately" to avoid partiality issues (see this earlier question of mine for details)....
- 19k
4
votes
1
answer
207
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On two centers of convex regions
Definition: A line segment with both end points on the boundary of a planar convex region $C$ is called a chord of $C$.
Consider any point $P$ within a given planar convex region $C$. From among all ...
- 5,493
3
votes
1
answer
196
views
"Almost geodesics" in Riemannian manifolds which cannot be loops
Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
- 21.1k
3
votes
0
answers
140
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Optimal intersections between planar convex regions
Here is an earlier discussion that could be related:
On comparing planar convex regions of equal perimeter and area
We are broadly interested in placing two given planar convex regions so that the ...
- 5,493
3
votes
1
answer
169
views
Criterion for visuality of hyperbolic spaces
I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."
Let $X$ be ...
- 97
2
votes
1
answer
125
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Curves of constant width that contain triangles
Wikipedia references: Curve of constant width,
Reuleaux polygon.
We record a pair of questions on the same lines as Smallest 3-ellipses that contain triangles.
Questions:
How does one find and ...
- 5,493
2
votes
0
answers
77
views
Another variant of the Malfatti problem
We try to add to A Variant of the Malfatti Problem
As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
- 5,493
2
votes
0
answers
109
views
More on shadows of 3D convex bodies
Ref: Shadows and planar sections of polyhedra
By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that ...
- 5,493
2
votes
1
answer
337
views
Union of Hamming balls
Let $V \subseteq \{0,1\}^n$, $\log|V| = k$. Consider
$V_r:= \bigcup_{x \in V} V_r(x)$, where $V_r(x)$ is a Hamming full-ball of radius $r$ and center $x$.
What is a lower bound for the cardinality ...
- 2,683