All Questions
1,357 questions with no upvoted or accepted answers
6
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176
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Approximating a ray with an integer lattice point
Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...
6
votes
0
answers
164
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Sets of points avoiding small angles
(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...
- 151k
6
votes
0
answers
476
views
Local isometry of complete length spaces that is not a covering map
Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...
- 2,934
6
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0
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369
views
Distance measures that preserve Pythagoras' theorem but break the triangle inequality
In information geometry, we can think of the Kullback-Leibler divergence as being "something like a squared distance."
The sense of this is that if we have three probability measures, $P$, $Q$ and $R$...
- 1,344
6
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0
answers
187
views
Isometric embedding of regular simplex into Riemannian manifold
Let $\{v_1,\cdots,v_k\}$ be the vertices of a regular $(k-1)$-simplex $\Delta(k,\ell)$, with a given metric such that the pairwise distance between the vertices is $\ell$.
Given a Riemannian ...
- 2,623
6
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0
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209
views
Stable norm on hyperbolic surfaces
For a hyperbolic surface $S$ and a homology class $h\in H_1(S)$ its stable norm is defined as $\lim_{n\to\infty}\frac{1}{n}l(nh)$, where $l(nh)$ means the minimal length among all closed geodesics ...
- 10.4k
6
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0
answers
191
views
Cut locus on a hypercube
Inspired by the question, "Shortest path connecting two opposite points on a cube":
Q. What does the cut locus with respect to one corner of a hypercube
in $\mathbb{R}^d$ look like?
"The cut ...
- 151k
6
votes
0
answers
384
views
Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
- 21.8k
6
votes
0
answers
281
views
Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
- 519
6
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0
answers
268
views
Bound on the determinant of a quadratic form restricted to a subspace
Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose:
The minimum of $Q(x,y)$ as $y$ varies ...
- 5,971
6
votes
0
answers
812
views
Limit of metric spaces
Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.
Assume ...
- 2,384
6
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answers
274
views
An inequality in cyclic polygon and tangential polygon
I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let $P$ ...
- 1,044
6
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0
answers
113
views
Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?
Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...
- 351
6
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0
answers
118
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Rational $d$-simplices
Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...
- 151k
6
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answers
303
views
Volume growth of balls
Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...
user1688
6
votes
0
answers
234
views
Tetrahedron incenter iteration tree
This is driven more by curiosity than by research,
but nevertheless may be of some interest.
Start with a regular tetrahedron $T$ with corners $(a,b,c,d)$,
and let $x$ be its incenter—the ...
- 151k
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
6
votes
0
answers
383
views
When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?
Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...
- 2,964
6
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0
answers
101
views
Unbalanced equipartitions
Let $K$ be a compact convex set in the plane.
Say that a perimeter-halving partition of $K$
is a partition of $K$
into two pieces by a chord (a segment with endpoints
on the boundary $\partial K$) ...
- 151k
6
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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 ...
- 151k
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
6
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0
answers
114
views
Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces
Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
- 161
6
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0
answers
189
views
Variations on a problem of S. Mazur
In problem 76 of the Scottish Book Mazur asked
Given a convex body $K$ in three-dimensional space and a point $o$ in its interior, consider the surface $S$ formed by all points $p$ such that the ...
- 13.5k
6
votes
0
answers
184
views
The Tangent Bundle of the Space of CR Structures on S^(2n+1)
Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
- 1,101
6
votes
0
answers
389
views
A conjecture of Thurston and possibly Weeks too
What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
- 1,131
6
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0
answers
260
views
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
0
answers
271
views
Families of triangulations of polygons in the plane
Let $P$ be a polygon in the plane. An "efficient" triangulation of $P$ is one that introduces no new vertices. We require that all introduced edges be straight and inside $P$. Every polygon in the ...
- 1,625
6
votes
0
answers
237
views
Generalization of the non-existence of a monostatic planar body
Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...
- 5,971
6
votes
0
answers
491
views
Minimum solid angle and aspect ratio of an $n$-simplex
In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.
In two ...
- 5,327
6
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0
answers
387
views
Local minimum from directional derivatives in the space of convex bodies
I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
- 5,971
6
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0
answers
302
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degenerating surface II
In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
- 914
6
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346
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Enriched Categories: Metric Spaces, Monoidal Endofunctors and Lipschitz-Continuous Maps.
In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:
Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every ...
- 3,249
6
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0
answers
442
views
Geometric meaning of gamma sets
First some notation:
Let $\mathscr{F}_* $ be the category of finite pointed sets and pointed maps between them. Then $\Gamma^{op}$ is the full subcategory of $\mathscr{F}_* $ with objects the sets $k_+...
- 998
6
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0
answers
2k
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Geometric Proof that Fubini-Study Metric is Round
The Fubini-Study metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round ...
- 159
6
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0
answers
176
views
Spaces with the thin tetrahedra property
I read a comment about the $\delta$-thin tetrahedra property of a space.
It basically means, that if you choose any four points in this space, connect them by geodesics, and fill each triangle with a ...
- 11.1k
6
votes
1
answer
896
views
Flat norm metrizes the weak* topology
I've come across the following statement in literature (without proof or reference) about the flat norm of currents
$$
F(T) = \sup \{ T(\omega) : \omega \in D^k(U), |\omega(x)| \leq 1, |d\omega(x)| \...
6
votes
1
answer
254
views
Triangulations of convex surfaces
Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$.
It is easy to see ...
- 7,149
5
votes
0
answers
78
views
Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
5
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0
answers
127
views
Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?
We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
- 24.9k
5
votes
0
answers
137
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Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
- 351
5
votes
0
answers
184
views
Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
- 3,507
5
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0
answers
146
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What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
- 193
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0
answers
75
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Integral over quotient of discrete group
Let $Y$ be a proper metric space. By a lattice we mean a discontinuous group of isometries $\Gamma$ with compact quotient $Y/\Gamma$. You may also assume that $\Gamma$ acts freely. Suppose we are ...
user473423
5
votes
0
answers
158
views
Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?
$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
- 51
5
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0
answers
270
views
Barycenter maps that are "submultiplicative" with respect to group actions
Background and notation
For a set $X$, we denote $\mathcal{P} (X)$ to be the finitely supported measures on $X$, i.e., $\nu \in \mathcal{P} (X)$ is of the form
$$\nu = \sum_{i=1}^n a_i \delta_{x_i},$...
- 1,163
5
votes
0
answers
120
views
Is an equilateral triangle constructible in a Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
- 41.8k
5
votes
0
answers
213
views
Does every smooth surface contain three congruent curves?
Let $M \subset \mathbb R^3$ be a smooth, connected, closed surface. We say a family of smooth curves on $M$ are congruent if each of them can be mapped to any other by a isometry of the ambient ...
- 6,155
5
votes
0
answers
171
views
Length metrics on covering spaces
This is a question (Exercise 3.30(2)) in the book `Metric spaces of non-positive curvature' written by Bridson and Haefliger.
In the book, there is the following proposition (Proposition 3.28)
Let $p:\...
- 439
5
votes
0
answers
177
views
Tiling with triangles of same circumradius and inradius
Consider a pair of positive real numbers $r$ and $R$ with $r<R/2$. Then we can form infinitely many triangles all with circumradius $R$ and inradius $r$.
For any such pair, the resulting triangles ...
- 5,979
5
votes
0
answers
74
views
Concentration bound on additive functions with constraints
Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...
- 107