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5
votes
1answer
419 views

A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{R}^n : \langle y, x \...
14
votes
2answers
270 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 (...
3
votes
1answer
178 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 0....
30
votes
5answers
882 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
111 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
90 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
0answers
142 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
2answers
419 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 A*\...
4
votes
1answer
169 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
262 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 ...
8
votes
3answers
856 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 ...
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 ...
3
votes
1answer
344 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
67 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 $p_2$...
1
vote
0answers
137 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
336 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 ...
6
votes
2answers
189 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 i,j,k\le4}\...
3
votes
1answer
222 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
68 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 $(n-1)$-...
7
votes
2answers
630 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 $M$...
4
votes
6answers
386 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 ...
4
votes
3answers
550 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
5answers
533 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
219 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
264 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 ...
8
votes
1answer
2k 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
264 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 non-...
1
vote
1answer
469 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 [...
15
votes
0answers
503 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
251 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
1answer
320 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
248 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
383 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
177 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
154 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
1answer
356 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
0answers
210 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
1answer
266 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
1answer
337 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
1answer
511 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
1answer
171 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
1answer
308 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
1answer
333 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 ...
26
votes
2answers
992 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
1answer
239 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
1answer
191 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
3answers
762 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
0answers
221 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 $\mathbb{R}^n$. ...
1
vote
1answer
242 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 $\...