**3**

votes

**1**answer

205 views

### Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...

**3**

votes

**0**answers

87 views

### Surfaces with curvature $\leq K$ are of bounded integral curvature

One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$
is the ...

**1**

vote

**1**answer

192 views

### Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$
I came across the following generalization of this equality:
$$\sqrt{1-k^2} {\rm ...

**3**

votes

**0**answers

76 views

### A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each
of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.)
Think of $T$ as a physical, rigid ...

**3**

votes

**1**answer

117 views

### Does this squared distance functional have a unique critical point on geodesically convex manifolds?

Let $M$ be a Riemannian manifold with distance function $d$, $C \subset M$ a geodesically convex set, $a=(a_i)_{i=1}^n \in C^n$, $W \in \mathbb{R}_{\geq 0}^{n \times n}$ and $J\colon C^n \rightarrow ...

**3**

votes

**0**answers

83 views

### Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...

**7**

votes

**3**answers

356 views

### The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...

**5**

votes

**0**answers

181 views

### Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...

**5**

votes

**1**answer

221 views

### Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear,
no four cocircular—such that all interpoint distances are rational.
Guy, Richard. Unsolved Problems in ...

**6**

votes

**1**answer

127 views

### Translative packing constant strictly larger than lattice packing constant

Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...

**4**

votes

**3**answers

272 views

### Peeling a polygonal vegetable

When you peel a vegetable, such as a potato or a cucumber, you usually remove its head, then contiually remove parts of its skin, until you remain with the pulp alone. I would like to formalize this ...

**34**

votes

**3**answers

1k views

### What polygons can be shrunk into themselves?

Let's call a polygon $P$ shrinkable if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the ...

**8**

votes

**1**answer

255 views

### Dubins car shortest paths: Decidable?

A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...

**3**

votes

**1**answer

237 views

### Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume ...

**6**

votes

**0**answers

117 views

### CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...

**30**

votes

**2**answers

611 views

### what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...

**6**

votes

**4**answers

364 views

### What can be said of the structure of a metric space without isosceles triangles?

This is a question that has been bothering me in the back of my head for quite some time.
Suppose we have a metric space $X$ with metric $\mathrm{d}$. By an isosceles triangle we mean a tuple of ...

**7**

votes

**2**answers

124 views

### Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take ...

**4**

votes

**0**answers

80 views

### Nice minimal embeddings of large finite groups into compact Riemannian manifolds

The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...

**15**

votes

**1**answer

268 views

### Higher dimensional generalization of: Any quadrilateral tiles the plane?

Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., ...

**18**

votes

**2**answers

557 views

### Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...

**11**

votes

**1**answer

318 views

### Simple, closed geodesics in $\mathbb{S}^3$ manifold

Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., ...

**4**

votes

**0**answers

123 views

### Circumscribing simplex to convex body?

Q.
Does every (perhaps smooth) compact convex body $K$ in $\mathbb{R}^d$ admit a
circumscribing simplex, each facet of which touches (shares a point with) $K$?
How about a circumscribing ...

**3**

votes

**1**answer

124 views

### Covering points with a shortest lattice spiral

Let $S$ be a finite set of lattice points in $\mathbb{Z}^2$.
My question is, roughly:
Q. How can a shortest lattice spiral that passes through
every point of $S$ be found?
A lattice spiral (my ...

**3**

votes

**1**answer

115 views

### Lipschitz function with somewhere dense image

Let $Q=[-1,1]^2$ denote the unit square and let $f:Q\to Q$ be a Lipschitz function such that for any ball $B(a,r)\subset Q$ with radius $r$, the width of the image $f(B(a,r))$ is at least $cr$ for ...

**8**

votes

**1**answer

528 views

### Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...

**4**

votes

**2**answers

113 views

### Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...

**9**

votes

**1**answer

258 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**9**

votes

**4**answers

556 views

### A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
Families of similar shapes, such as squares, are closed in this metric. Also more general ...

**1**

vote

**1**answer

141 views

### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where ...

**2**

votes

**0**answers

166 views

### Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group ...

**10**

votes

**1**answer

241 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**1**

vote

**0**answers

88 views

### Quadrilaterals from a Unit Stick

This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):
Suppose one are given an ...

**0**

votes

**1**answer

125 views

### Two questions about convex subsets of Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a ...

**5**

votes

**0**answers

112 views

### Star shaped sets with a midpoint

Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the ...

**7**

votes

**1**answer

98 views

### Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of ...

**2**

votes

**2**answers

114 views

### Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...

**3**

votes

**1**answer

120 views

### Local geodesics in uniquely geodesic spaces

A while ago I asked this
question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.
Suppose $Y$ is a proper length space, where every pair of points $x,y\in ...

**2**

votes

**1**answer

71 views

### Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...

**10**

votes

**3**answers

349 views

### Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates ...

**5**

votes

**2**answers

235 views

### Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a shortest halving curve if
(a) it partitions the ...

**15**

votes

**0**answers

458 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

**2**

votes

**0**answers

98 views

### Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...

**3**

votes

**2**answers

213 views

### Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...

**-3**

votes

**2**answers

149 views

### Hexagon Formed by connecting Trisections of triangle sides [closed]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...

**10**

votes

**2**answers

512 views

### Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the ...

**27**

votes

**2**answers

868 views

### Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...

**3**

votes

**1**answer

121 views

### Are there isomeasure simplices?

Say that two polyhedra in $\mathbb{R}^3$ have isomeasures
(my terminology) if they have:
the same volume,
the same surface area,
the same sum of all edge lengths,
and the same number of vertices.
The ...

**23**

votes

**2**answers

710 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**3**

votes

**0**answers

87 views

### maximum sum of angles between $n$ lines

Take $n$ lines in $\mathbb{R}^d$ (not necessary different, and all passing through the origin, though this is not important). What is maximal possible sum of angles between them for given $n$ and $d$? ...