Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
692 questions
7
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2
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Cone unfolding of space curves
There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ...
- 7,149
7
votes
1
answer
648
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Maximal volume of a simplex inscribed in a spherical cap
Let $B_n$ be the $n$-dimensional unit ball, and $B_n(\varepsilon)$ be the spherical cap with height $\varepsilon$ I am interested in the quantity
$$\Gamma:=\sup_{\Delta:\textrm{ inscribed simplex in }...
- 599
7
votes
1
answer
582
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Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann's approach)
This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, ...
- 272
7
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4
answers
1k
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On discrete version of curve shortening flow
One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...
7
votes
1
answer
497
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Is there a bicyclic irregular pentagon in integers?
Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well?
If it does ...
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7
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2
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392
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Convex deltahedra in higher dimensions
There are eight convex polyhedra whose faces are equilateral triangles, so-called
deltahedra:
(Image from here)
Q. Have the equivalent higher-dimensional ...
- 151k
7
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1
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373
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Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
- 6,741
7
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205
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Lattice radial-step (ratchet) spirals
(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...
- 151k
7
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1
answer
399
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Objects whose morphisms are Lipschitz maps
I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz").
The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $...
- 250
7
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2
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180
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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 ...
- 10.6k
6
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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-...
- 5,979
6
votes
2
answers
3k
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Finding the convex combination of vertices which yields an inner point of a polytope
Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
- 243
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On planar sections of 3D convex bodies
Consider the space of planar sections of any given convex 3D body.
Basic Question: What is the lower bound for the ratio
$$\frac{\text{area of section of greatest perimeter}}
{\text{area of section of ...
- 5,979
6
votes
1
answer
311
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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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0
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219
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How big a box can you wrap with a given polygon?
Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
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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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0
answers
260
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Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?
Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
- 13.5k
6
votes
2
answers
497
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Average distance of the mean of $n$ random complex numbers in a unit disc
Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
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6
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1
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761
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Checking if one polytope is contained in another
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...
- 491
6
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1
answer
408
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Rectangles in rectangles and $(b^2-a^2)^2\le (ax-by)^2+(bx-ay)^2$
When does an $a\times b$ rectangle fit inside an $x\times y$ rectangle? I have an algebraic condition which I can diagram geometrically, and I'd like a good geometric argument.
Assume $0<a<b$, $...
user44143
6
votes
0
answers
217
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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 \...
user6671
6
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3
answers
2k
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Euler's rotation theorem revisited - Elementary geometric proofs
This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE.
Euler's Rotation Theorem, proved ...
6
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5
answers
4k
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Formulas for equidistant curves
I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first ...
- 351
6
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0
answers
182
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Factorization of metric space-valued maps through vector-valued Sobolev spaces
Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
1
answer
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
6
votes
2
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575
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Which groups are doubling?
A metric space $(M,d)$ is doubling if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric ...
- 6,652
6
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2
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544
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On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
6
votes
1
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273
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Locally compact + two-point homogeneous => Riemannian
A metric space $M$ is called two-point homogeneous if for any pair of points $(p,q)$ in $M$ any distance preserving map $f\colon\{p,q\}\to M$ can be extended to an isometry $\bar f\colon M\to M$.
The ...
- 45k
6
votes
1
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168
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Are $\varepsilon$-connected components dense?
Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...
- 41.8k
6
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549
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Volume doubling, uniform Poincaré, counterexample
The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
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6
votes
1
answer
212
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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
6
votes
1
answer
524
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Enlarging a tetrahedron with integer edge lengths
Given a tetrahedron with all edges having integer length, is it always possible to increase all of the edge lengths by one?
More precisely: Let $P_1, P_2, P_3, P_4$ be four distinct non-coplanar ...
- 856
6
votes
1
answer
1k
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Compact manifolds locally bi-Lipschitz to Euclidean space
I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
- 35.5k
6
votes
2
answers
189
views
Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π.
Given a convex polygon, how does one algorithmically find the point (...
- 5,979
6
votes
2
answers
215
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Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
- 5,979
6
votes
1
answer
228
views
Does this iterated Delaunay triangulation process always "explode"?
Let $P$ be a set of three noncollinear points in $\mathbb{R}^2$.
Iteratively form the
Delaunay triangulation
$\cal T$ of $P$, and then
augment $P$ by the circumcircle centers of all triangles in $\...
- 151k
6
votes
2
answers
729
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Rationality of translation lengths in hyperbolic groups
Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...
- 619
6
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4
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708
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Surface dissection of regular tetrahedron to cube
Does anyone know what is the fewest-piece
dissection
of the surface of
a regular tetrahedron to the surface of a cube (of the same area)?
It is well-known that the volume of a regular ...
- 151k
6
votes
1
answer
513
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The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher
According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
- 245
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1
answer
237
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m-point-homogeneous, but not (m+1)-point-homogeneous
It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ ...
- 45k
6
votes
2
answers
217
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Untangling entwined rigid chains in 3-space
I am interested in exploring the degree of "tangledness"
of two rigid chains in space.
A polygonal chain is a simple (non-self-intersecting) path
of segments in
$\mathbb{R}^3$, viewed as a rigid body. ...
- 151k
6
votes
1
answer
281
views
Convex sets in Alexandrov spaces
Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary?
Definition of a convex subset: $A\subseteq X$ ...
- 377
5
votes
1
answer
383
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cover and hide with squares
I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...
- 3,549
5
votes
3
answers
479
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Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?
Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:
If we rotate $C$ around $p$...
- 289
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1
answer
292
views
All-set-homogeneous spaces
This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property?
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...
- 45k
5
votes
0
answers
508
views
Longest simple path through hypercube corners
This is a variation on a previously answered question,
Longest path through hypercube corners.
Here I am seeking the longest simple (non-self-intersecting) path through
the unit hypercube's vertices,
...
- 151k
5
votes
1
answer
1k
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Smoothness of the square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. The distance between two points is the infimum of the lengths of the curves which join the points. Consider the square of the distance function
$d^2\...
- 59
5
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0
answers
1k
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"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
- 151k
5
votes
3
answers
683
views
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
- 13.6k
5
votes
0
answers
266
views
Throwing darts at a barn and putting a bullseye around them in higher dimensions
Let $X \in \mathbb R^d$ be a large domain (a ball of radius $r$ for $r$ large should suffice)
Let $B$ be a ball of radius $1$.
Consider the ratio
$$ \frac{ \left| \left\{ x_1,\dots,x_n \in X \mid ...
- 148k