**1**

vote

**1**answer

72 views

### 4D Duoprisms based on nonconvex polygons

A duoprism is a polytope
that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$).
Four-dimensional duoprisms in particular have been studied:
$$P \times Q = \{ ...

**0**

votes

**1**answer

70 views

### Existence of shortest paths in complete Alexandrov spaces

Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...

**13**

votes

**2**answers

266 views

### Can any simplex shadow-project to a regular simplex?

Every triangle $A$ can be oriented in $\mathbb{R}^3$
so that its orthogonal projection (shadow) onto the $xy$-plane is an
equilateral triangle $Q$:
...

**1**

vote

**0**answers

159 views

### Products between metrics in a product of manifolds

In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...

**17**

votes

**5**answers

419 views

### How many unit simplices are needed to cover a unit $n$-cube?

The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.
...

**4**

votes

**2**answers

343 views

### Terminology for metrics?

For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...

**3**

votes

**1**answer

183 views

### Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...

**3**

votes

**0**answers

74 views

### On the modulus of convexity of mixed-norm $\ell_{p_1,p_2}$ spaces

Let $\ell_{p_1,p_2}=(\mathbb{R}^{m\times n},\|\cdot\|_{p_1,p_2})$ be the space of $m\times n$ matrices endowed with the mixed-norm
$$ \|X\|_{p_1,p_2} = \left( \sum_{j=1}^n \left( \sum_{i=1}^m ...

**0**

votes

**0**answers

75 views

### Largest ball with fixed center in a a convex region

Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...

**3**

votes

**2**answers

621 views

### What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
...

**0**

votes

**0**answers

116 views

### How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$.
It seems that such a set cannot be very large and that its ...

**2**

votes

**1**answer

121 views

### Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact).
Question 1. Is it true that every point of $X$ has a ...

**13**

votes

**2**answers

657 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

109 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

85 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

112 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 ...

**5**

votes

**3**answers

255 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

787 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

101 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

134 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

217 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

83 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

72 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

97 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

416 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

121 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

126 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

276 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

89 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

145 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

189 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

131 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

154 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

168 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

323 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

155 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

178 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

154 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

284 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

139 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

153 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

71 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

212 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

345 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
...

**2**

votes

**1**answer

86 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 ...

**8**

votes

**0**answers

168 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

304 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

110 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

266 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

235 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 ...