All Questions
1,357 questions with no upvoted or accepted answers
7
votes
0
answers
305
views
Generalizing Gromov Hausdorff distance using Vietoris topology
There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...
- 1,630
7
votes
0
answers
209
views
Stabbing disks in space, or: Galactic alignment
I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...
- 151k
7
votes
0
answers
119
views
Approximating manifolds with boundary by closed ones
Fix numbers $n\in \mathbb{N},d>0,k\in\mathbb{R}$. Do there exist numbers $N\in\mathbb{N},K\in\mathbb{R}$ depending on $n,d,k$ only with the following property:
For any compact smooth Riemannian $n$...
- 21.8k
7
votes
0
answers
904
views
Geometry of level sets of a convex function
EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the ...
- 21.8k
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
7
votes
0
answers
318
views
Status of an open question in Artin's "Geometric Algebra"
In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).
The ...
- 501
7
votes
0
answers
1k
views
Closed-form solution of a linear programming question
Among all the probability matrices
\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \...
7
votes
0
answers
284
views
Shortest path to inspect a polyhedron
This is a variant of two as-yet unsolved MO questions cited below.
Let $P$ be a closed polyhedron in $\mathbb{R}^3$.
The task is to find a shortest path $\sigma$ on the surface of $P$ from which
all ...
- 151k
7
votes
0
answers
154
views
Connectedness of cones in the boundary of a 1-ended hyperbolic group
Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
- 611
7
votes
0
answers
477
views
Gromov's compactness theorem for manifolds with boundary
The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
- 21.8k
7
votes
0
answers
156
views
Thales Style Level Sets
Encouraged by Joseph O'Rourke ( and inspired by the discussion at
Thales' semicircle theorem in higher dimensions ), I ask about level sets in three
dimensional space occuring from considering ...
- 2,158
7
votes
0
answers
277
views
Reversing shortest paths among unit disks
Twas the night before Christmas, and throughout M.O.
Not a question was posted, not even by Joe.
Well, let me remedy that. :-)
Let the plane contain a number of ...
- 151k
7
votes
0
answers
251
views
Equiareal shapes in $\mathbb{R}^d$
There was quite a bit of work on the so-called
equichordal problem throughout the 20th century, to decide if some plane convex
curve could have two equichordal points.
A point is equichordal for a ...
- 151k
7
votes
0
answers
205
views
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
votes
0
answers
1k
views
What is known about the area of the symmetric Pythagorean tree?
What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to ...
- 205
7
votes
0
answers
292
views
Minimal spanning tree of a point set in the unit square, under an unusual distance function
For two points $x$, $y \in [0,1]^2$, let their distance be $d(x,y) := \|x-y\|_2^2$ (i.e. the usual distance, squared). Technically, this is a semimetric, as it does not satisfy the triangle inequality....
- 1,318
7
votes
0
answers
177
views
Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric?
Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others under an isometry?
- 700
7
votes
0
answers
152
views
Almost Isodiametric Sets
Hi,
The isodiametric inequality tells us that, of all sets of diameter $r$, the one with the largest Lebesgue measure is the ball of radius $r/2$ - and this holds regardless of norm. Let $\tau(r)$ be ...
- 131
7
votes
0
answers
323
views
Erlangen program carried out explicitely?
I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations.
What I mean by "carrying out Erlangen program" is picking a specific group (say SO(...
- 71
7
votes
0
answers
669
views
Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
- 151k
7
votes
0
answers
208
views
How do metrics behave under joining along a manifold embedded in the boundary?
How do metrics behave under joining along a manifold embedded in the boundary?
This is, more-or-less, part of Problem 4.66 in Kirby's List:
Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
- 1,897
6
votes
0
answers
197
views
What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
6
votes
0
answers
172
views
Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
- 565
6
votes
0
answers
184
views
When is a distance space dominated by a metric space?
A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
- 193
6
votes
0
answers
153
views
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 ${\equiv}\subseteq X^2\times X^2$ ...
- 41.8k
6
votes
0
answers
68
views
Vector algebra in a Tarski space
By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
- 41.8k
6
votes
0
answers
111
views
Does the Segment-Circle Axiom imply the Circle-Circle Axiom in a non-Euclidean Tarski plane?
By a Tarski plane I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
- 41.8k
6
votes
0
answers
189
views
What is a non-smooth connection?
Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
- 63.9k
6
votes
0
answers
121
views
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
74
views
Roundest polyhedra: how well can we bound the edge count of their faces?
By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
- 3,612
6
votes
0
answers
182
views
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
$$
\int_{x\in X}\,d(y_0,f(x)...
- 5,405
6
votes
0
answers
219
views
How big a box can you wrap with a given polygon?
Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
- 5,979
6
votes
0
answers
134
views
Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ ...
- 45k
6
votes
0
answers
132
views
Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
6
votes
0
answers
320
views
Does this plane geometry theorem have a name (well-known)?
Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
- 1,776
6
votes
0
answers
112
views
Which $n$-gons of diameter 1 maximize the moment of inertia?
Background: Among convex plane $n$-gons of unit diameter, we can try to achieve:
the largest area. (This is called the biggest little polygon with $n$ sides; for $n$ odd, the regular polygon on $n$ ...
- 5,979
6
votes
1
answer
604
views
When is the cut locus a finite tree?
Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
- 5,038
6
votes
1
answer
489
views
What inequalities for convex sets are known since the work of Scott and Awyong?
In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
- 3,068
6
votes
0
answers
264
views
Odd Steinhaus problem for finite sets
Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $A\subset \mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points....
- 18.7k
6
votes
0
answers
62
views
Continuity of embeddings and systole as you vary a metric
Let $M$ be a smooth compact manifold and let $R(M)$ be the set of Riemannian metrics on $M$, topologized with the $C^\infty$ topology (viewing a metric as a section of an appropriate bundle).
I have ...
- 61
6
votes
0
answers
101
views
Shortest path on Riemannian manifold with boundary
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume
$$II_{\partial M}(...
- 21.8k
6
votes
0
answers
247
views
An extension of Erdos' distinct distances problem based on circles of various radii
Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the ...
- 24.7k
6
votes
0
answers
130
views
ultrametric Rademacher theorem
The classic Rademacher theorem roughly states that Lipschitz continuous functions are almost everywhere differentiable. However, there are well-known ultrametric counterexamples, see Kobliz's classic ...
- 500
6
votes
0
answers
217
views
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 $X$:
$$d(v,w) = 1-\frac{2 \...
user6671
6
votes
0
answers
176
views
Area-preserving map of punctured disk to itself
If $D_r = \{v\in \mathbb{R}^2 : 0 \lt |v| \lt r\}$, consider the map $f_r: D_r \to D_r$ given by:
$$f_r(x,y) = \frac{\sqrt{r^2-x^2-y^2}}{\sqrt{x^2+y^2}}\left(-y,x\right)$$
Geometrically, $f_r(v) \...
- 2,902
6
votes
0
answers
109
views
"Moduli space" of isotropic convex bodies?
A lot of questions in convex geometry revolve around the geometry of isotropic convex bodies in $\mathbb{R^n}$.
To my knowledge there is no, or very little study of a space such as :
$$C_n = \{...
- 245
6
votes
0
answers
132
views
Is there any work in topological data analysis on something like "Voronoi complexes"?
Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...
- 15.4k
6
votes
0
answers
367
views
Adjoint of the Hodge de Rham star operator under the integral pairing
Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration ...
- 5,824
6
votes
0
answers
1k
views
How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?
Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
- 13.4k
6
votes
0
answers
92
views
What quantum groups admit quantum topography space structure?
Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
- 361