Deprecated; please use a more specific tag.

**3**

votes

**1**answer

137 views

### “Trapping” of discs after random sequential adsorption

Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx ...

**27**

votes

**4**answers

751 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**4**

votes

**2**answers

101 views

### Simulating random sequential adsorption in reverse

Please consider two processes:
Process 1 - I simulate random sequential adsorption of discs on the unit square in the continuum limit, randomly selecting real number coordinates and rejecting the ...

**0**

votes

**1**answer

77 views

### Random Sequential Adsorption of Discs on a Plane - What is the best known lowerbound for the number of circles (of some radius $r$) guaranteed to fit on $[0, 1]^2$?

Imagine I perform a random sequential adsorption (RSA) simulation for circles or discs of some radius $r \leq 1$ in $[0, 1]^2$ (I am open to changing this geometry to the unit circle). As a function ...

**0**

votes

**0**answers

128 views

### Relation between the index of (the sum of) lattices in euclidean space, and their orthogonal complement.

I am trying to figure out something concerning the index of lattices.
The question came about after reading the paper of W.Fulton and B.Sturmfels, ("Intersection theorey on toric varieties"). To ...

**3**

votes

**2**answers

273 views

### The probability distribution for vertex degree in a unit disc graph generated from random points on a plane

Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx ...

**1**

vote

**1**answer

173 views

### What kinds of manifolds admit concave boundary?

We can find many examples of smooth Riemannian manifolds with boundaries whose boundaries are convex. But it seems to me I know no any example of smooth Riemannian manifold with concave boundary. So ...

**1**

vote

**0**answers

273 views

### question regarding $S^2-D$ being equidecomposable with $S^2$ in Banach-Tarski paradox [closed]

In the Wikipedia proof of Banach-Tarski paradox (http://en.wikipedia.org/wiki/Banach-Tarski_paradox#Some_details.2C_fleshed_out), there is a stuff about $S^2-D$ being equidecomposable with $S^2$ as ...

**3**

votes

**1**answer

293 views

### Reverse Gauss's Circle Problem

Gauss's Circle Problem [1] is to find the number of lattice points inside the boundary of a circle with a given radius and center at the origin. I'm interested in the "reversed" version of this ...

**4**

votes

**1**answer

133 views

### Nontrivial lower bounds on Cheeger inequalities for Markov chains

For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...

**4**

votes

**1**answer

251 views

### A question of compactness in the geometry of numbers

Given a star body $S \subset \mathbb{R}^n$ with the origin as interior point, the critical determinant of $S$---usually denoted as $\Delta(S)$---is the infimum of the determinants of all lattices ...

**-1**

votes

**1**answer

92 views

### How to define an “anisotropic vector” for a given object?

Dear experts,
I am looking for a way to define an "alignment vector" (or anisotropy or orientation vector?) for a given geometrical object. I am not sure how to put this into correct technical terms, ...

**8**

votes

**3**answers

529 views

### Solid angles of a tetrahedron

This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the ...

**1**

vote

**0**answers

53 views

### Lorentzian isoparametric hypersurfaces of ads

A hypersurface of a pseudo-Riemannian manifold is said to be isoparametric if its shape operator has the same characteristic polynomial at all points. Xiao has classified Lorentzian isoparametric ...

**1**

vote

**0**answers

52 views

### Sandwich a l-p ball between Simplexes

Let $U \subseteq \mathbf{R}^d$ be the unit ball with respect to the $l_p$-norm. I would like find the minimal $m=m(d)$ such that that there is a simplex $S$ with $S\subseteq U \subseteq mS$.
For the ...

**2**

votes

**0**answers

81 views

### Largest subsets of quadrics consisting of “nonorthogonal” vectors

Assume we have an $A$-module $M$, and a quadratic form $q : M \to A$. Recall that it means that
1) $q (a m) = a^2 q (m)$ for all $a \in A$ and $m \in M$, and
2) $B_q (x, y) := q (x + y) - q (x) - q ...

**3**

votes

**1**answer

246 views

### Are all null curves of a Lorentzian metric extrema?

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of
$$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$
The so-called "null curves" are ...

**1**

vote

**1**answer

49 views

### Uniformly sampling the solution space for points where the free termini of two rays, anchored at 3-space points, can intersect

I have two rays, one of length $L_1$ and one of length $L_2$. I anchor these rays, each at one end, on the 3-space points $p_1$ and $p_2$. Assuming that the Euclidean distance between $p_1$ and ...

**1**

vote

**0**answers

133 views

### Relationship between stabilizers of a general point and a boundary point

Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...

**3**

votes

**1**answer

274 views

### The right conformal map to make a certain picture

This is a follow-up to a question I asked a year ago, which was helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
I am trying to come up with a way to make a picture of an ...

**8**

votes

**0**answers

174 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

172 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

194 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

421 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

296 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 ...

**0**

votes

**1**answer

568 views

### Optimal fitting of spheres in a cylinder.

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

**3**

votes

**3**answers

351 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 ...

**1**

vote

**0**answers

151 views

### Find minimum area ellipse enclosing a set of ellipses, all centered at the origin

Thanks to Will Jagy for answering my similar question re: two ellipses.
Find minimum area ellipse which encloses two ellipses
I had initially naively thought I could apply this solution to a set of ...

**11**

votes

**5**answers

506 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

200 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

208 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 ...

**5**

votes

**2**answers

414 views

### Find minimum area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...

**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

222 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

278 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

417 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

230 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

228 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

229 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

249 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

158 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

145 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

341 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

193 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

242 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

286 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

368 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 ...