**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 ...

**1**

vote

**0**answers

75 views

### Alexandrov spaces satisty $BE(K,N)$ and $BE(K,\infty)$?

Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi ...

**0**

votes

**1**answer

126 views

### harmonic forms with respect to different metrics

Given a smooth manifold M with two different metric, we can consider U and V, the spaces of harmonic k-forms on M with respect to the first metric and second metric respectively. The question is ...

**6**

votes

**1**answer

545 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
$X_{n}$ is a regular CW complex of constant local dimension $n$.
$X_{n}$ is of finite type, ...

**6**

votes

**2**answers

406 views

### Is there a 3d equivalent of this picture?

This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
The picture below is the image of a regular ...

**4**

votes

**2**answers

174 views

### point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...

**9**

votes

**1**answer

255 views

### Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction?
The answer does ...

**7**

votes

**0**answers

112 views

### Can two random graphs be metrically embedded into one another?

Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal ...

**3**

votes

**0**answers

435 views

### Is compass and straight edge geometry complete?

Euclid's first three postulates are the basis of compass and straight edge constructions which are as complex as arithmetic.
The constructions themselves may be expressed as a formula with each of ...

**2**

votes

**1**answer

113 views

### Are characteristic maps of CW complexes Lipschitz up to homotopy?

Let us consider a finite CW complex $X=\cup X_j$ with a given metric, compatible with the topology (maybe a reasonable one coming from some embedding into some $\mathbb{R}^n$).
The characteristic ...

**9**

votes

**2**answers

431 views

### Efficiently determine if convex hull contains the unit ball

Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have ...

**3**

votes

**1**answer

64 views

### Points in general position on a small grid

A point set $P$ is said to be embedded in $\mathbf{Z}^2$ in general position, if no three points lie on a common line. Assume that $|P|=n$, I am interested in the smallest $k \times k$ integer grid in ...

**3**

votes

**3**answers

319 views

### Intersection of Polyhedra

I'm writing a collision detection algorithm, and so far I've been using Joseph O'Rourke's book "Computational Geometry in C" as reference. It outlines an algorithm to determine whether a point is ...

**7**

votes

**1**answer

174 views

### Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?

**18**

votes

**2**answers

851 views

### Probability that a convex shape contains the unit ball

This probability problem seems interesting and I don't know if it has been solved before.
If you pick $n$ points uniformly at random from the surface of a $d$ dimensional sphere of radius $r>1$ ...

**9**

votes

**2**answers

299 views

### Rational points on circular spirals

Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...

**7**

votes

**2**answers

281 views

### Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set
$$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$
Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by
...

**2**

votes

**1**answer

130 views

### Upper bounds on the worst-case traveling salesman tours in the unit square

The paper [1] proves that, if we place $N$ points in the unit square, then the length $\ell$ of the euclidean TSP tour of those points must satisfy $$\ell \leq \sqrt{2N} + 7/4~~.$$ I'm wondering, can ...

**7**

votes

**2**answers

254 views

### Minor theorems of Pappus and Desargues in “old school” geometry?

My question concerns the dependence relations between the minor theorem of Pappus which, following Heyting, I will denote by $P_9$, and (one of the) minor theorems of Desargues, $D_9$.
$P_9$ states ...

**8**

votes

**1**answer

271 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 ...

**6**

votes

**0**answers

178 views

### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not ...

**4**

votes

**0**answers

132 views

### $L_1$ and $L_\infty$ Voronoi diagrams and tropical geometry: Connection?

I just realized that there is a visual similarity between Voronoi diagrams in
the $L_1$ and $L_\infty$ metrics (two images below)
Left: O'Rourke, "Computing Relative Neighborhood graph ...

**3**

votes

**2**answers

155 views

### A question about maximizing the volume of a particular kind of convex set

Let C be a compact convex subset of 3-dimensional Euclidean space E(3) whose interior is non-empty and whose diameter is d. What is the largest volume that C can have if every subset of C that is a ...

**5**

votes

**1**answer

116 views

### name for a polytope constructed from a system of linear equations?

To a system of inhomogeneous linear equations, one can associate a polytope, as follows. Let $A\in\mathbb{R}^{m\times n}$,
$b\in\mathbb{R}^m$ and $V=\{x\mid Ax=b, \text{support of $x$ ...

**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 ...

**4**

votes

**1**answer

377 views

### Question on a concrete example of n points

Can anyone give a concrete example of n points in the unit square (for instance, n runs from 3 through a large number) that can be generated by the algorithm here or any other algorithm or any ...

**12**

votes

**2**answers

397 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 ...

**0**

votes

**0**answers

45 views

### curvature of subset of Alexandrov spaces

If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$ds_M^2 = d{t^2} + \exp \left( {2t} \right)ds_N^2$$ N is a submanifold of M. Then by Gauss-equation, we can prove $Ric\left( N ...

**7**

votes

**1**answer

154 views

### Can a tangle of arcs interlock in plane?

This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here:
Can a (finite) collection of disjoint circle arcs in ...

**4**

votes

**1**answer

179 views

### n-simplex in an intersection of n balls

Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...

**5**

votes

**3**answers

945 views

### Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it on a book a few years ago. Though I've tried to solve it, I'm facing difficulty.
Question: You know on a plane there is an ...

**2**

votes

**0**answers

296 views

### Partitioning the Projective Plane

Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$.
Side Note: If there are more standard definitions for any of the ideas presented here, ...

**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 ...

**3**

votes

**4**answers

340 views

### Conditions for when an off-centre ellipsoid fits inside the unit ball

An ellipsoid $E$ has centre $\vec{c}=(c_1,c_2,c_3)$ and semiaxes $t_1$, $t_2$ and $t_3$ aligned with the $x$, $y$ and $z$ axes. What are the necessary and sufficient conditions on $\vec{c}$, $t_1$, ...

**7**

votes

**1**answer

344 views

### Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...

**2**

votes

**1**answer

110 views

### Amalgmated free product of hyperbolic groups with one malnormal and one virtual factor is hyperbolic?

A possible formulation of the Bestvina-Feighn theorem is as follows (taken from here):
Combination Theorem (Bestvina & Feighn): If $H$ is a malnormal subgroup of hyperbolic groups $G_1, G_2$, ...

**0**

votes

**1**answer

69 views

### Is a $CD(K,\infty)$ space a length space?

Let $(X,d)$ be a complete and separable metric space endowed with a nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray}
\mu(B(x,r))<\infty,\quad\mbox{for every }x\in ...

**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

135 views

### Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain

Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?

**5**

votes

**1**answer

122 views

### 2-layer tilings with a center-of-gravity constraint

I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...

**3**

votes

**1**answer

259 views

### Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...

**7**

votes

**0**answers

366 views

### Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...

**4**

votes

**1**answer

149 views

### Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.

**6**

votes

**1**answer

148 views

### Proof of a statement from Steele's “Probability theory and combinatorial optimization”

I am reading "Probability theory and combinatorial optimization" by J.M. Steele and am hung up on a statement made in Section 2.2 of Chapter 2, "Easy size bounds", in which it is stated (paraphrasing ...

**2**

votes

**2**answers

193 views

### Geodesics in Graphs:Shortest Paths vs Going as Straight Ahead as Possible

According to Wikipedia http://en.wikipedia.org/wiki/Geodesic, a geodesic " is a generalization of the notion of a "straight line" to "curved spaces " and further " In the presence of an affine ...

**4**

votes

**3**answers

472 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**1**

vote

**1**answer

79 views

### Expected length of a certain kind of nearest-neighbor graph

Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...

**7**

votes

**2**answers

263 views

### Line-preserving bijection of ${\mathbb{R}}^n$ onto itself

If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ not assumed), must $f$ be affinity?
Assuming continuity would certainly ...

**2**

votes

**0**answers

63 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 ...

**6**

votes

**2**answers

220 views

### How close can closed geodesics be?

A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such ...