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14
votes
2answers
245 views

Random rings linked into one component?

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (tilted) randomly ...
6
votes
2answers
827 views

What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
2
votes
0answers
169 views

Finding Center of Union of Circles [closed]

Hello, I have a set of N circles on a 2-D plane. These N circles all contain the same point (coordinates unknown) so there is a common union between all the circles. How could I go about finding ...
3
votes
1answer
132 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
4answers
748 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
2answers
99 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
1answer
75 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 ...
4
votes
1answer
179 views

Monotonicity of Loewner ellipsoid?

Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$? I have just finished a proving a ...
0
votes
0answers
127 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 ...
2
votes
2answers
258 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
1answer
171 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
0answers
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
1answer
291 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
1answer
131 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
1answer
250 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
1answer
82 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
3answers
500 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
0answers
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
0answers
51 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
0answers
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 ...
0
votes
0answers
196 views

About the parallel transport and choice of connection

Thought Experiment Consider a 2-sphere, $S^2$, and let $p$ be a point at the equator. Case 1 Let us parallel transport a vector, $V$ from $p$ using the recipe: Move one unit of length East. Move ...
3
votes
1answer
217 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
1answer
48 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
0answers
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
1answer
268 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
0answers
173 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
2answers
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
1answer
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
1answer
191 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
0answers
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
2answers
405 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
5answers
290 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
1answer
516 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
3answers
324 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
0answers
147 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
5answers
502 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
0answers
199 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
2answers
193 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
2answers
394 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
1answer
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
1answer
218 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
1answer
250 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 ...
11
votes
0answers
392 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
1answer
228 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
0answers
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
1answer
219 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
0answers
225 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
2answers
236 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
0answers
157 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
0answers
141 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 ...