All Questions
4,825 questions
0
votes
0
answers
29
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,465
2
votes
1
answer
136
views
Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
- 1,776
1
vote
0
answers
28
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
-1
votes
0
answers
41
views
Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
-3
votes
0
answers
47
views
Proof AG = 2EF in an Isosceles Right Triangle [closed]
In an isosceles right triangle ABC with angle ACB = 90 degrees and angle CAB = angle ABC, let point G lie inside triangle ABC. In the isosceles right triangle CGE, where angle CGE = 90 degrees and CG =...
- 1
10
votes
1
answer
156
views
For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
- 263
0
votes
1
answer
90
views
How to calculate the maximum dimensions of a rectangle inside two concentric circles? [closed]
If I have a rectangle ABCD such that A and B touch two points of the outer circle and CD's touches one point of the inner circle, how could the maximum dimensions of the rectangle be calculated?
...
- 17
2
votes
0
answers
67
views
Pólya's orchard problem among Gaussian primes
Quoting myself from an earlier post:
Pólya's orchard problem asks for which radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of ...
- 151k
1
vote
0
answers
66
views
Quasi-geodesics in Alexandrov spaces
I am trying to understand the notion of quasi-geodesic in Alexandrov space with curvature bounded below following the Perelman-Petrunin paper. I have two questions:
Is it true that the shortest ...
- 21.8k
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
2
votes
1
answer
168
views
Ratio of inscribed/circumscribed ellipsoids: geometrical proof?
Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
- 52.3k
1
vote
0
answers
37
views
Metric entropy of an ellipsoid
Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map.
Consider the function
$$
H(T) := \log M(TB_2^d, B_2^d),
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$....
- 460
9
votes
0
answers
143
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
1
vote
0
answers
31
views
Cut locus of linear isometric action quotients
Given a compact group $G\leq \operatorname{O}(d)$ of linear isometries on $\mathbb R^d$, equip its quotient $\mathbb R^d/G$ with the canonical orbital metric.
I am curious about the following. Is ...
- 71
-1
votes
0
answers
64
views
Axes of symmetry and symmetry group of the tangent cone to an open, connected, convex subset of the Euclidean space
Given a closed convex set $K\subset \mathbb{R}^d$ and a point $x\in K$ the tangent cone to $K$ at $x$ is defined by
\begin{equation}
T_xK:=\overline{\{v\in \mathbb{R}^d: \exists \lambda \geq 0 \text{ ...
- 1,465
0
votes
0
answers
72
views
Reflections of Voronoi diagrams
I wonder if something similar to the following fact is known, and I would greatly appreciate any references.
Let $t_1, t_2, \ldots, t_N$ be unit vectors in $\mathbb{R}^n$.
Let $S$ denote the unit ...
- 1
2
votes
1
answer
108
views
Discrete isoperimetric inequality involving the diameter of an n-gon
I am interested in discrete isoperimetric-type inequalities that allow one to bound the perimeter of an $n$-gon from above (as opposed to below, as in the classical case when one bounds the perimeter ...
- 1,625
0
votes
1
answer
67
views
Metric for measuring linearity of finite set of points in $R^2$
Suppose one has $n > 2$ points in $R^2$, and one wants to measure "how linear" they are.
I want a metric such that (a) if all the points are in fact on the same line, the metric gives 1, (...
0
votes
0
answers
42
views
Reference request: in Alexandrov geometry gradient flows preserve extremal subsets
It is mentioned in literature that in Alexandrov geometry gradient flows of semi-concave functions preserve each extremal subset.
I am looking for a proof of this fact.
- 21.8k
2
votes
0
answers
49
views
Riemannian metrics realizable as hypersurfaces both in Euclidean and spherical spaces
I am interested in smooth Riemannian metrics on $n$-sphere, $n\geq 3$, which can be imbedded isometrically both to $n+1$-dimensional Euclidean space and $n+1$-dimensional standard sphere of radius $r$....
- 21.8k
5
votes
1
answer
161
views
I believe that all facets of a Voronoi-cell of a lattice are centerally symmetric. Is my argument correct? Is this true?
So let $L$ be a full dimensional lattice in $\mathbb{R}^{n}$. Then the Voronoi-cell of the lattice are precisely the points in $\mathbb{R}^{n}$ that are at least as close to the origin, as to any ...
1
vote
1
answer
181
views
Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
4
votes
0
answers
97
views
What is the best way to subdivide a simplex?
Let $\Delta^k$ be the $k$-simplex, embedded in $\mathbb{R}^{k+1}$ in the usual way so that all edges have length $\sqrt{2}$. For $k\leq 2$, there are obvious ways to subdivide $\Delta^k$ into $2^k$ ...
- 56.9k
3
votes
0
answers
49
views
Transport map to lower dimension?
Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$.
Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
- 171
2
votes
0
answers
93
views
Understanding Gromov's metric measure space
Sorry for organized the question badly. My supervisor forced me to read chapter $3\frac 12$ of the reputed book Metric structures for riemannian and non-riemannian spaces written by Mikhail Gromov, ...
- 21
0
votes
0
answers
36
views
Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
3
votes
0
answers
96
views
Filling radius of Lens spaces
This is a question concerning Gromov's filling radius, i.e., the radius of a neighborhood of a Riemannian manifold (embedded in its Banach space of $L^\infty$-functions) at which the fundamental class ...
- 517
6
votes
1
answer
346
views
Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
- 93
2
votes
0
answers
152
views
Isoperimetric inequality for Kähler manifolds
I am interested in the following form of isoperimetric inequality for Kähler Manifolds (for example unit ball $B^n\subset \mathbb{C}^n$ with Bergman metric). It should say something like this: if $F$ ...
- 37
9
votes
0
answers
240
views
Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 128k
25
votes
2
answers
2k
views
Is there a continuous partition of space into circles?
Question 1. Is there a continuous partition of space $\mathbb{R}^3$ into circles?
I strongly suspect not.
It is well-known by diverse arguments that space can be partitioned into circles. There is an ...
- 236k
21
votes
0
answers
270
views
The "stained glass window problem": Draw many random chords in a circle; which kind of polygon ($3$-gon, $4$-gon, etc.) occupies the most total area?
Draw $n$ random chords in a circle, where each chord connects two independent uniformly random points on the circle.
As $n\to\infty$, which kind of polygon (triangle, quadrilateral, pentagon, etc.) ...
- 3,507
2
votes
0
answers
125
views
How does a conformal transformation affect the frame bundle metric of that manifold?
Suppose I have a metric $g_{\mu\nu}$ over an n-dimensional smooth orientable Riemannian manifold $M$. We then utilize Cartans repere mobile (moving frames) to define oriented orthonormal frames $e^{a}=...
- 250
0
votes
0
answers
29
views
Stable gap-less packing of a box with boxes
define a box packing as gap-less if
all inner boxes have disjoint interior
the sum of volumes of the inner box equals that of the outer box
the sum of the extents of the inner boxes in each principal ...
- 13.2k
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157
10
votes
5
answers
738
views
Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
- 82.6k
2
votes
0
answers
30
views
Grid point density with the most (Demaine) neighbors
Here is a random distribution of points on a $12\cdot 12$ grid illustrating Demaine neighbors (as you can see, it can happen that a horizontal or vertical has no points at all - that doesn't cause ...
- 4,829
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
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
1
vote
2
answers
188
views
Non-compact surfaces with non-negative Gauss curvature
Is there a topological classification of non-compact complete connected 2-dimensional Riemannian manifolds with non-negative Gauss curvature?
- 21.8k
4
votes
1
answer
97
views
Inner regularity property of covering number of metric spaces
Let $(X,d)$ be a complete metric space and $n\in\mathbb N$. Suppose that every finite subset $F\subset X$ can be covered by $n$ closed balls of $X$ (that is, $N(Y,d,1)\le n$, in terms of covering ...
- 60.5k
4
votes
0
answers
124
views
Is this monoid generated from lines in $\mathbb{R}^d$ cancellative and torsion-free?
Let $L^d = \{\mathbb{R}v \ : \ v \in \mathbb{R}^d \setminus \{0\}\}$ denote the set of lines through the origin of the real vector space $\mathbb{R}^d$. I am interested in a commutative monoid ...
- 41
0
votes
0
answers
25
views
Is there a name for a spanner graph that only considers distance to a root node?
A $t$-spanner graph of a set of points $\{p_i\}$ in the plane is a graph $G = (V, E)$ such that for any pair of vertices $p_i, p_j \in V$, the shortest path distance $d_G(p_i, p_j)$ in $G$ is at most $...
- 4,049
7
votes
1
answer
290
views
Is it necessarily true that the maximal section of a centrally symmetric convex body is always bigger than its minimal projection?
I hope everyone is doing well.
Let $K \subset \mathbb{R}^n$ be a centrally symmetric convex body $(K = -K)$. Denote by $K \mid H$ the orthogonal projection of $K$ onto $H$, where $H$ is an $n - 1$ ...
- 83
10
votes
0
answers
160
views
Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
- 2,934
4
votes
1
answer
96
views
Sequence of 2-cylinders converging to a segment in the Gromov-Hausdorff metric
Let $\{C_i\}_{i=1}^\infty$ be a sequence of (compact) 2-dimensional cylinders with smooth Riemannian metrics with Gauss curvature at least $-1$ and geodesically convex boundary (equivalently, the ...
- 21.8k
1
vote
0
answers
33
views
Collapse of Moebius bands with bounded below Gauss curvature and convex boundary
Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the ...
- 21.8k
0
votes
0
answers
176
views
How to find a configuration of lines
In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
- 61