**13**

votes

**2**answers

649 views

### (Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...

**7**

votes

**0**answers

97 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**

votes

**0**answers

81 views

### Is there any counterpart to Thales' semicircle theorem in higher dimensions?

It was established by TMA, @WillSawin, and @DouglasZare, in their responses to
the MO question,
"Thales' semicircle theorem in higher dimensions,"
that the natural generalization of Thales' semicircle ...

**2**

votes

**0**answers

109 views

### Intersecting balls with convex regions and a bisector thereof

This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...

**4**

votes

**2**answers

166 views

### How can dimension depend on the point?

Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...

**15**

votes

**5**answers

753 views

### Thales' semicircle theorem in higher dimensions

Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle.
Q1. Does a cone with apex on a hemisphere and encompassing the circular base
have a solid angle ...

**2**

votes

**0**answers

94 views

### Geometry of rings and semi-rings

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.
One direction would be the following. Consider $\mathbb{N}$ (with the ...

**6**

votes

**1**answer

128 views

### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...

**3**

votes

**2**answers

202 views

### distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that
$$
\int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...

**2**

votes

**1**answer

77 views

### Control of the metric in isothermal coordinates

Suppose you have a riemannian surface $(\Sigma,g)$, and an open simply-connected set $U \subset \Sigma$. You know that you can find isothermal coordinates - that is a map $\varphi : U \rightarrow D$ ...

**1**

vote

**0**answers

67 views

### A version of isotone projection cones

We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...

**2**

votes

**1**answer

86 views

### distance formula of warped products

Given a warped product, I want to compute the ditance of any two points. First get the equation for the geodesic, then compute the length of the geodesic.
Consider the two-dimensional surface
$$
...

**38**

votes

**1**answer

401 views

### Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$

Let $f : \mathbb{R} \longmapsto \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a line in ...

**2**

votes

**1**answer

120 views

### hyperbolic metrics

Let $D_1\subset D_2$ be simply connected domains in the complex plane. Let $\lambda_1$ and
$\lambda_2$ be the corresponding hyperbolic (Poincare) metrics. It seems intuitive to me that
$\lambda_2$ is ...

**2**

votes

**0**answers

119 views

### Dynamics of electrons on a sphere

Suppose one place $n$ electrons closely surrounding the north pole of a sphere, forming
a perfect planar regular $n$-gon:
Q1.
What will happen if the ...

**2**

votes

**1**answer

265 views

### Perimeter of ellipse: Combination of two geometries

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter ...

**2**

votes

**0**answers

82 views

### Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...

**0**

votes

**2**answers

132 views

### Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?

Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?

**2**

votes

**1**answer

182 views

### Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic:
The ...

**1**

vote

**0**answers

84 views

### Is there a projective metric on a projective space induced by a p-norm?

A projective metric on a projective space is a metric on the underlying set such that shortest path with respect to this metric are parts of entire projective straight lines.
The 2-norm induces the ...

**2**

votes

**1**answer

145 views

### Closed geodesics that cross one another frequently

Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$.
$S$ has at least three simple (non-self-intersecting), closed geodesics by
a theorem of Lyusternik and Shnirel'man.
Alternatively, ...

**1**

vote

**2**answers

122 views

### Curves similarity metric [closed]

I am working on an optical character recognition algorithm that takes vector data (i.e. polylines) as input rather than raster picture. E.g., we have N polyline samples, and when certain polyline is ...

**4**

votes

**3**answers

284 views

### Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...

**2**

votes

**1**answer

142 views

### length comparison on negatively curved surfaces

Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...

**7**

votes

**1**answer

152 views

### Upper bound for the number of integral points in a convex set

Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer
coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?

**2**

votes

**0**answers

121 views

### Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...

**2**

votes

**0**answers

268 views

### Good covering of a sphere

Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$.
We will be interested in covering this sphere with balls of radius $\rho < r$.
We know that there ...

**5**

votes

**2**answers

124 views

### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...

**3**

votes

**2**answers

126 views

### regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...

**4**

votes

**0**answers

64 views

### Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...

**6**

votes

**0**answers

182 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**14**

votes

**1**answer

315 views

### 3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the
tesseract, into 3D.1
These unfoldings (or "nets") are analogous to the 11 unfoldings of
the 3D cube into the plane.2
...

**1**

vote

**0**answers

47 views

### How often can a single length occur as a boundary distance?

Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$?
We can make any assumptions about the ...

**7**

votes

**0**answers

121 views

### Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...

**5**

votes

**1**answer

287 views

### Geodesics on manifolds with boundary

Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...

**2**

votes

**0**answers

105 views

### How far can I push a ball of radius r into the corner of a convex polytope in d-dimensions?

Let $K \subset \mathbb{R}^d$ be a convex polytope and let $B(x,r) = \{y \in \mathbb{R}^d \,:\, \Vert y-x \Vert_2 \leq r\}$ be the ball of radius $r$ centered at $x$.
For any radius $r \geq 0$ let ...

**2**

votes

**1**answer

230 views

### An identity for Futaki-Donaldson invariant

Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...

**2**

votes

**2**answers

230 views

### Length of non-horizontal curve

Let $M$ be a sub-Riemannian space.
Consider a smooth curve $\gamma:[0,1]\to M$ such that
$\dot\gamma(t)\not\in H_{\gamma(t)}$, where $H_{\gamma(t)}$ is the horizontal subbundle ( i.e. $\gamma$ is ...

**2**

votes

**1**answer

85 views

### Is there a name for the level-sets of the signed distance function to a set in a metric space?

$\newcommand \X {\mathcal{X}}$
$\newcommand \sd {d_{\rm sign}}$
Let $(\X, d)$ be a metric space and define the distance between a point $x \in \X$ and a set $S \subset X$ by $d(x,S) = \inf_{y \in S} ...

**9**

votes

**1**answer

147 views

### Optimal shape for stabbing balls in $\mathbb{R}^3$

I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region,
say, within a unit-radius sphere $S$.
I shoot a ray/path through $S$, hoping to ...

**4**

votes

**1**answer

119 views

### Distance function from a topological submanifold

Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can ...

**2**

votes

**1**answer

150 views

### Lorentzian metrics on the torus up to continuos deformations

Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.
Which ...

**1**

vote

**0**answers

106 views

### Virtually abelian centralizers

This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)
Anyway, I'm wondering what sort of groups have the ...

**2**

votes

**1**answer

88 views

### Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?

Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's ...

**0**

votes

**1**answer

189 views

### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...

**8**

votes

**0**answers

129 views

### Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...

**4**

votes

**0**answers

188 views

### How many points does 'the-most-point-contained-circle' contain at least?

Question : Given any $n$ distinct points $S$, consider the $\binom n2$ discs $D_{pq}$ formed by picking a pair of points $p,q$ from $S$ and using them as a diameter. For each disc $D_{pq}$, let ...

**15**

votes

**2**answers

413 views

### Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...

**7**

votes

**0**answers

106 views

### Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...

**7**

votes

**1**answer

242 views

### The Minkowski sum of two curves

Let $\gamma$ be a continuous curve in the complex plane without self-intersections and let $\lambda$ be a complex non-real number less than 1 in modulus. Put $\gamma'=\lambda\gamma$.
Question. Is it ...