# Tagged Questions

**4**

votes

**1**answer

182 views

### Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...

**0**

votes

**1**answer

75 views

### Constructing measures with support in a given set

I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...

**7**

votes

**1**answer

415 views

### Is this knot invariant already treated somewhere in the literature?

Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...

**2**

votes

**1**answer

103 views

### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

**3**

votes

**2**answers

295 views

### Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...

**13**

votes

**0**answers

380 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**4**

votes

**1**answer

135 views

### Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature

This is probably an easy question, but I don't know any Riemannian geometry and a literature search hasn't helped. Any help (e.g. providing a reference) would be greatly appreciated.
For a triangle ...

**6**

votes

**1**answer

95 views

### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant ...

**4**

votes

**1**answer

73 views

### Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...

**2**

votes

**0**answers

89 views

### Definition of self-dual polytope

Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...

**2**

votes

**1**answer

102 views

### Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...

**2**

votes

**1**answer

101 views

### Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$

A simple polygons in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by list the coordinates of it's vertices. I expect there might be something interesting to study ...

**2**

votes

**2**answers

240 views

### Sphere - Symmetry and Triangulation [closed]

The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...

**34**

votes

**1**answer

2k views

### Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**8**

votes

**1**answer

275 views

### Oloid and sphericon: rolling develops entire surface

Wikipedia says that,
"The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface."
Below are illustrations of ...

**2**

votes

**2**answers

1k views

### Who first realized that the shortest distance between two points is a straight line? [closed]

It's a bit like asking who invented the wheel, but perhaps there's something out there. This seems beyond the obvious crew: Euclid, Pythagoras, etc. Is there any evidence when this became common ...

**12**

votes

**2**answers

398 views

### Sets of evenly distributed points in the Euclidean plane

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can ...

**2**

votes

**2**answers

161 views

### Three questions concerning lattice points on sphere surfaces

Pardon my ignorance of this topic.
Q1.
In which dimensions $d$ is it the case that, for every natural number $n$,
there exists a sphere having exactly $n$ lattice points on it ...

**7**

votes

**0**answers

178 views

### Bi-spherical polyhedra

Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...

**2**

votes

**1**answer

209 views

### Passing C through a slot

Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?
i.e. which are the necessary and sufficient conditions for a planar compact set C to ...

**0**

votes

**2**answers

136 views

### Hales's fan associated with a polyhedron

In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in $\mathbb{R}^3$.
I will not define his notion of fan, but let his figure (p.137) serve as a definition:
...

**11**

votes

**0**answers

237 views

### Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?

In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' ...

**2**

votes

**1**answer

181 views

### 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 ...

**10**

votes

**1**answer

430 views

### Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
...

**4**

votes

**4**answers

516 views

### The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...

**4**

votes

**2**answers

142 views

### Comparing two Delaunay tessellations on a hyperbolic surface

Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb ...

**5**

votes

**1**answer

137 views

### Matching on sphere to create cycle with chords

Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like ...

**3**

votes

**1**answer

103 views

### Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...

**3**

votes

**0**answers

361 views

### N-balls covering n-balls

This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...

**4**

votes

**0**answers

92 views

### Metrized categories

Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let ...

**5**

votes

**1**answer

261 views

### Reference request: affine transforms + circle inversion?

This problem cropped up in the context of scale-insensitive methods for generating random variables.
Let $X=R^n \cup \{\infty\}$. Suppose we consider a set of transforms $\cal{T}$ from $X\rightarrow ...

**8**

votes

**0**answers

229 views

### Optimal inspection path on a sphere

Suppose you would like to "inspect" every point of a unit-radius
sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$
on $S$, but you can only see a distance $d$ from where you stand.
...

**0**

votes

**0**answers

114 views

### Optimal paintbrush geodesics

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$,
and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece
of a geodesic.
Define $\gamma(w)$ as all the points of $S$ within
a distance ...

**5**

votes

**2**answers

387 views

### For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...

**2**

votes

**2**answers

282 views

### Reference for ultrametric spaces

I have a research project involving ultrametric spaces, and there are some facts that I use but have a hard time finding explicitely in the literature, although I know that some of them are folklore ...

**6**

votes

**1**answer

612 views

### reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct:
Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...

**5**

votes

**3**answers

258 views

### Which metric spaces have this superposition property?

Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$.
So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the superposition property (my terminology)
...

**12**

votes

**1**answer

551 views

### The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...

**4**

votes

**0**answers

431 views

### Egg-ovoid rolling down an inclined plane

I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane,
for pedagogical reasons.
It is well-known folk lore that the shape of an egg prevents it from rolling away from
...

**1**

vote

**0**answers

139 views

### Compute generalized pentagram map

Hi,
(This is my first question on MathOverflow! :-)
Imagine you have a set of points $S = \{p_1, \ldots, p_n\}$ in $\mathbb{R}^d$, of which $t$ are "bad". I want to compute a "safe convex hull", ...

**8**

votes

**1**answer

318 views

### Isoperimetric inequality in complex hyperbolic space

Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space.
This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid
...

**14**

votes

**4**answers

1k views

### Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...

**11**

votes

**1**answer

307 views

### Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...

**18**

votes

**1**answer

2k views

### Proposed Counterexample to a Theorem of Differential Geometry (on Banach manifolds)

This question stems from Jeff Rubin's earlier MO question and a follow-up that I posted.
The former recalls the following result proved by both Serge Lang (Fundamentals of Differential Geometry, ...

**13**

votes

**1**answer

322 views

### Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...

**21**

votes

**2**answers

1k views

### Non-regular Connected Hausdorff Banach Manifold

After reading this MO post, I am wondering:
Is every (connected) Hausdorff Banach manifold a regular space?
Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...

**9**

votes

**3**answers

432 views

### Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the ...

**4**

votes

**0**answers

539 views

### Curves of constant curvature on an ellipsoid

It is not difficult to see that the curves of constant geodesic curvature on a geometric sphere
are all circles: simple, closed curves that are geometric circles lying in a plane:
...

**6**

votes

**2**answers

337 views

### Source for: Geodesics in CAT(0) spaces

I am seeking a good introductory reference that could lead to an understanding of
the properties of geodesics in
complete CAT(0) metric spaces.
I am especially interested in learning the differences ...

**3**

votes

**4**answers

477 views

### Hermitian Christoffel Symbols

Does anyone know of some good references for computing Christoffel symbols for Hermitian metrics?
A quick Google search turns up this. The following formula appears on page 4:
$$\Gamma_{AB}^C = ...