Deprecated; please use a more specific tag.

**8**

votes

**0**answers

183 views

### Does there always exist a self dual polytope that contains a given polytope contained in its dual?

Suppose a polytope $P$ is contained in its dual polytope $\tilde{P}$. Does there always exist a polytope $Q$ that contains $P$ and is self dual $Q=\tilde{Q}$? Is there any bound on the minimal number ...

**6**

votes

**2**answers

173 views

### Untangling entwined rigid chains in 3-space

I am interested in exploring the degree of "tangledness"
of two rigid chains in space.
A polygonal chain is a simple (non-self-intersecting) path
of segments in
$\mathbb{R}^3$, viewed as a rigid body. ...

**1**

vote

**1**answer

421 views

### Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is
$$\inf_{\begin{array}{c}
v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\
v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right)
\end{array}}\max_{0\le ...

**3**

votes

**1**answer

205 views

### What properties does generalized Delaunay triangulation have?

Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...

**2**

votes

**0**answers

63 views

### Are there a group of mappings from (n-1)-dim space to an (n-1)-sphere guaranteeing the orthogonality of images?

Hello, everyone.
As we know that in an $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a continuous bijective mapping from a subset $V^{n-1}\subseteq\mathbb{R}^{n-1}$ to a unit ...

**6**

votes

**2**answers

476 views

### Relating curvature and torsion of a connection to those of a curve

I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion.
I know that an affine connection $A$ on an $n$-dimensional manifold ...

**4**

votes

**5**answers

313 views

### Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...

**-1**

votes

**1**answer

647 views

### Optimal fitting of spheres in a cylinder.

how to find the minimum height and width of a cylinder containing n identical spheres?

**4**

votes

**3**answers

416 views

### On duality on finite projective planes

Hey Everyone!
In nearly all (if not all) projective geometry texts I have bumped into the following theorem:
"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for ...

**11**

votes

**5**answers

522 views

### To what extent does trajectory determine gravity sources?

Suppose one has in-hand an accurate time-space trajectory in $\mathbb{R}^3$ of a (small) body,
say an asteroid or satellite—effectively a point.
To what extent does this trajectory determine the ...

**1**

vote

**0**answers

207 views

### Pencils of circles and Liouville's theorem

Is there any relation (maybe implicit) between the conformal geometry in the space of circles and spheres and the study of harmonic functions?
In the original question I was musing whether the ...

**1**

vote

**2**answers

226 views

### Reference question: Poncelet theorem

A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in ...

**7**

votes

**1**answer

1k views

### Geometric interpretations of matrix inverses

$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point ...

**2**

votes

**1**answer

234 views

### another diameter-perimeter-area inequality

Recently I learnt that $$
\inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no ...

**1**

vote

**1**answer

334 views

### The largest circle that encloses no points on a plane with points placed at $N$ random coordinates

I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in ...

**12**

votes

**0**answers

439 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**7**

votes

**1**answer

238 views

### Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...

**2**

votes

**0**answers

96 views

### Isometric decomposition

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others an isometry?

**2**

votes

**1**answer

251 views

### A quantitative version of Straszewicz's theorem?

Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an exposed point of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are ...

**4**

votes

**0**answers

237 views

### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
...

**0**

votes

**2**answers

282 views

### Enlcosing a set of ellipses within one ellipse

Hello,
Is there an algorithm that takes in a set of ellipses and gives back and ellipse that encloses the set?

**4**

votes

**0**answers

164 views

### Convex bodies with symmetric shadows.

Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry.
This is a classic result ...

**4**

votes

**0**answers

146 views

### Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?

Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...

**3**

votes

**1**answer

343 views

### What is the expected value for this

If there are $8$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the open interval
$\left(0,1\right)$, what is the expected largest size of ...

**6**

votes

**0**answers

202 views

### Families of triangulations of polygons in the plane

Let $P$ be a polygon in the plane. An "efficient" triangulation of $P$ is one that introduces no new vertices. We require that all introduced edges be straight and inside $P$. Every polygon in the ...

**0**

votes

**1**answer

251 views

### AC and Euclidean Geometry [closed]

It there any relation between the axiom of choice and Euclidean Geometry ??
I mean what are the known statements, theorems or results in euclidean geometry that are dependent on AC ?? (this question ...

**2**

votes

**1**answer

299 views

### Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form.
Some background may be ...

**2**

votes

**1**answer

410 views

### Minimizing the Perimeter of a polyomino

Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...

**1**

vote

**1**answer

166 views

### Orbits of Product Lie Groups Action

Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...

**11**

votes

**1**answer

295 views

### Polyominoes with double contact

Here is a problem which arose from an earlier question. I'll change the terminology but not the question: A polyomino is a region with a connected interior made by joining one or more unit squares ...

**1**

vote

**1**answer

302 views

### Frustrating the number of possible common edges between two connected components composed of square Penrose tiles

Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...

**24**

votes

**2**answers

821 views

### The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.
...

**4**

votes

**1**answer

210 views

### Extreme rays in the cone of (semi)metrics

How many extreme rays are there on the polytopal cone formed by all semimetrics on a set with $n$ elements?
Some background. Given a set $X$ with $n$ elements, the set of all semimetrics
$d:X \times ...

**2**

votes

**1**answer

189 views

### What is the Sequence that Maximizes this Distance?

I have posted this question here without answer. Maybe I can get some light here.
Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where ...

**14**

votes

**3**answers

656 views

### Smallest square to wrap a cylinder

Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle)
of height $h$ and radius $r$.
Here wrap is the natural sense of covering the surface area of the cylinder ...

**7**

votes

**0**answers

207 views

### From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine.
Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...

**0**

votes

**1**answer

220 views

### Smooth maps transverse to a foliation

Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$ ($\dim S < \dim N$). Let $\mathfrak S$ be a foliation of $S$. We say that a map between $M$ and $N$ is transverse to ...

**2**

votes

**2**answers

227 views

### What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of
subsets in a subdivision of the set of $30$ points into subsets such
that all the points in each subset are on the boundary of the ...

**29**

votes

**3**answers

1k views

### “Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...

**8**

votes

**1**answer

405 views

### Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$.
Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if
there is a continuous rotation of $P$ such that its orthogonal-projection
shadows ...

**1**

vote

**1**answer

273 views

### Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$
Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...

**0**

votes

**1**answer

215 views

### Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...

**2**

votes

**2**answers

413 views

### Convex upper bound on a linear-fractional function

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ...

**2**

votes

**1**answer

296 views

### Manhattan distance vs. absorption time on an unbounded integer lattice

Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...

**5**

votes

**2**answers

127 views

### Abstract characterization of polygonizations

Consider a polygonization of the plane by convex polygons of a given minimal size that meet edge-to-edge and vertex-to-vertex.
What's the “official” name of such a polygonization?
Such ...

**4**

votes

**3**answers

272 views

### lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post)
Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes ...

**11**

votes

**2**answers

998 views

### There are two points on the Earth's surface that … ?

At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...

**2**

votes

**2**answers

563 views

### Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry"， an English version ...

**1**

vote

**1**answer

324 views

### Leray Spectral Sequence

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$.
Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$
be a generic fiber that is a ...

**13**

votes

**1**answer

341 views

### The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...