All Questions
718 questions
4
votes
1
answer
385
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Divergence of geodesics in mapping class groups
I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...
- 619
4
votes
0
answers
123
views
From a given triangle, to cut 2 mutually congruent convex pieces that together 'use' maximum area of the triangle
Two planar regions are congruent if one can be made to perfectly coincide with the other by translation, rotation or reflection (flipping over).
The Problem: Given a triangular region T, how will we cut ...
- 5,979
4
votes
2
answers
476
views
Geodesic transformations of the complex projective plane
Are there non-trivial diffeomorphisms (i.e., different from isometries) of the complex projective plane that map geodesics (for the canonical Riemannian metric) to geodesics?
Same question for all ...
- 13.5k
4
votes
3
answers
376
views
Proof that lifts of geodesics are quasi-geodesics (relatively hyperbolic groups)
$\DeclareMathOperator\Cay{Cay}$Suppose $G$ is a relatively hyperbolic group with peripheral subgroups $P_1,P_2,\dots, P_n$, and suppose $\mathcal{S}$ is a finite generating set for $G$. Let $X=\Cay(G,\...
- 421
4
votes
1
answer
254
views
On the Lipschitz constant outside the stretch set
Let $f: \mathbb R^n \to \mathbb R^m$ be a Lipschitz map. We define the local Lipschitz constant $Lf$ of $f$ at $x \in \mathbb R^n$ by
$$Lf(x) := \lim_{r \to 0_+} \text{Lip}(f, B_r (x)),$$
where $\text{...
- 6,155
4
votes
2
answers
846
views
Fundamental polygons with infinite pairwise identifications
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
...
- 151k
4
votes
2
answers
811
views
Is it possible to sample uniformly on the surface of a high-dimensional polytope?
There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes.
Are there any methods that sample uniformly on the surface of a high-dimensional ...
- 143
4
votes
0
answers
232
views
Illuminating a just-barely irrational polygon
As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Let $P$ be a rational polygon.
Then for ...
- 151k
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
4
votes
1
answer
303
views
On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
- 5,979
4
votes
1
answer
159
views
Extending a metric in a bi-Lipschitz way
Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz ...
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
4
votes
0
answers
73
views
circular disk in a convex domain
This question was asked on https://math.stackexchange.com/questions/4014421/circular-disk-in-a-convex-domain1 but got no reply
Let $\partial \mathcal D$ be the smooth boundary of a convex domain $\...
- 179
4
votes
0
answers
132
views
Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
- 13.6k
4
votes
0
answers
140
views
Do the interiors of curves of constant width admit "linear" measures?
Concise question: In two dimensions, do all shapes of constant width admit a measure over their interior such that for any two parallel lines intersecting the shape, the area between them under the ...
- 41
4
votes
2
answers
312
views
Which convex pentagon gives least packing density?
Among all convex pentagons, does the regular pentagon give least packing density?
Further question: For each $n > 6$, is the regular $n$-gon the minimum of packing density?
An analogous question ...
- 5,979
4
votes
2
answers
575
views
Routh's theorem in three dimensions
The most well known case of Routh's triangle theorem is:
If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, ...
- 371
4
votes
2
answers
282
views
Fixed points on spherical buildings
A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an ...
- 12.9k
4
votes
1
answer
343
views
A question about bisecting plane convex sets
Suppose that S is a compact convex subset of the Euclidean plane E whose interior is non-empty.
If p is a point of E such that every straight line in E which passes through p bisects the area
of S, is ...
- 6,215
4
votes
1
answer
266
views
A closed chain of $2n+1$-gon around $2n+1$-points
I posed a generalization of Theorem 3.2 In my paper
Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;...
- 1,776
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
2
answers
446
views
About Euclidean distances
$\newcommand\R{\mathbb R}$Let $0<d_1<\cdots<d_k<\infty$ and let $m_1,\dots,m_k$ be any integers $\ge1$. Let $n:=m_1+\dots+m_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.
Do then ...
- 128k
4
votes
1
answer
522
views
Can every manifold be represented as a quotient
My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post.
Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
- 5,405
4
votes
1
answer
250
views
Zoll Flat Finsler tori and convex bodies on a starry night
The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the "...
- 13.5k
4
votes
0
answers
144
views
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)....
- 21.1k
4
votes
2
answers
2k
views
Simplified knapsack problem
There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
- 41
4
votes
2
answers
2k
views
Breaking a rectangle into smaller rectangles with small diagonals
Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
- 4,049
4
votes
0
answers
321
views
Extendability of isometries of convex surfaces
Let $K$ be a 3-dimensional convex compact subset either in $\mathbb{R}^3$ or in the 3-dimensional real hyperbolic space $\mathbb{H}^3$. Consider its boundary $\partial K$ equipped with the inner (path)...
- 21.8k
4
votes
1
answer
700
views
Embedding graphs into hyperbolic spaces
Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?)
I don't ...
- 617
4
votes
1
answer
482
views
Dirichlet polyhedra for hyperbolic manifolds
Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume ...
- 9,237
4
votes
0
answers
904
views
Packing space by cones: Translates best?
Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...
- 151k
4
votes
0
answers
756
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
4
votes
1
answer
321
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 ...
- 18.7k
4
votes
1
answer
423
views
Fixed points of finite order isometries of metric spaces
I would like to show the following:
Let $X$ be a complete metric space that is uniquely geodesic (i.e. each two distinct points are connected by a unique geodesic segment) and $\phi\colon X\to X$ an ...
- 41
4
votes
1
answer
891
views
Basic result in semi-infinite linear programming
Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
- 4,049
4
votes
2
answers
309
views
Finitely isometrically persistent metric spaces
The goal of this question is to develop further the discussion
initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...
- 4,878
4
votes
2
answers
4k
views
Curves similarity metric [closed]
I am working on an optical character recognition algorithm that takes vector data (i.e. polylines) as input rather than raster picture. E.g., we have N polyline samples, and when certain polyline is ...
- 151
4
votes
1
answer
251
views
A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius
Does there exist a function $\tau(\varepsilon)=\tau(\varepsilon,n,K,\mu)$ such that $\lim_{\varepsilon\to +0}\tau(\varepsilon)=0$ and for any $n$-dimensional complete Riemannian manifold $M^n$ with ...
- 21.8k
4
votes
2
answers
320
views
Inequality from a point in plane to a triangle OR Inequality on a quadrilateral
If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{...
- 55
4
votes
1
answer
131
views
Any visualization software for the intrinsic metric of a convex polyhedron?
I'd like to find a visual simulation of what it would be like to 'live' in a polyhedron with the intrinsic, piecewise-Euclidean length metric. Of course, to make it easier to visualize, I'd prefer to ...
- 3,359
4
votes
0
answers
446
views
Computing the volume of intersection between a ball and a box
$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
4
votes
1
answer
422
views
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
For $k = 2$ the answer is obvious since we can always place circles so that every one of them ...
- 63
4
votes
0
answers
216
views
Is each metric continuum $\ell_p$-chain connected?
This problem was motivated by the MO problems:
"Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".
...
- 41.8k
4
votes
1
answer
204
views
Reference: Packing under translation is in NP
I am looking for a reference for a result that I am aware of.
Let me describe the result.
Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP
time, if $p_1,\ldots,p_n$ can be ...
- 479
4
votes
3
answers
927
views
Lower bound for the normal injectivity radius
Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a ...
- 131
4
votes
0
answers
101
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 $f(x_0)=x_1$...
- 12.4k
3
votes
2
answers
517
views
Threading pinholes in the wall of cylinder to pass through an internal coordinate
Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...
3
votes
1
answer
358
views
Dirichlet form on Riemannian manifold is tight?
$M$ is an $n$-dimensional Riemannian manifold. Consider the Dirichlet form $$\varepsilon \left( {u,v} \right) = \int_M {\left\langle {\nabla u,\nabla v} \right\rangle }, \quad u ,v \in {W^{1,2}}\left( ...
- 689
3
votes
1
answer
173
views
Parameterizing the space of convex quadrilaterals
If $P=\mathbb{R}^2$ is the plane, is there a continuous surjection from $P^4$ to the space of convex quadrilaterals?
Specifically, I'm looking for a continuous $f:P^4\to P^4$ such that:
[convexity] ...
user44143
3
votes
2
answers
498
views
Totally geodesic submanifold of codimension 1
This question is inspired by question in reference.
Question : If $M$ is a simply connected closed Riemannian manifold of nonnegative sectional curvature, then there is a totally geodesic ...
- 1,100