5
votes
1answer
152 views

Matching on sphere to create cycle with chords

Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through the center of $S$, in such a way that no pair of chords intersect:       I would like ...
3
votes
1answer
116 views

Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov]. Is the same true ...
4
votes
1answer
500 views

Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane there are known algorithms for finding the largest empty circle ($LEC$) of the set of points. The $LEC$ problem is stated in this way: find a $LEC$ whose ...
7
votes
0answers
264 views

Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane, with some density $\rho$ per unit area. View the points as disks of radius zero. Now the radii $r$ of all disks grows ...
8
votes
1answer
299 views

Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedron for which this ...
5
votes
1answer
264 views

Reference request: affine transforms + circle inversion?

This problem cropped up in the context of scale-insensitive methods for generating random variables. Let $X=R^n \cup \{\infty\}$. Suppose we consider a set of transforms $\cal{T}$ from $X\rightarrow ...
3
votes
1answer
214 views

Sums of uniformly random vectors from the $n$-dimensional unit ball

I'm interested in some instances of the following problem. Let $n \geq 2$, and suppose we draw $k \geq 2$ vectors $v_1, \dots, v_k$ uniformly at random from the $n$-dimensional ball of radius $1$, ...
8
votes
0answers
261 views

Optimal inspection path on a sphere

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand. ...
0
votes
0answers
123 views

Optimal paintbrush geodesics

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$, and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece of a geodesic. Define $\gamma(w)$ as all the points of $S$ within a distance ...
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 ...
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 ...
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 ...
11
votes
2answers
936 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 ...
12
votes
1answer
297 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 ...
8
votes
2answers
882 views

About MF Atiyah and R Bott's 1983 paper

I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
11
votes
1answer
328 views

Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere $S$ in $\mathbb{R}^3$, sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$, with $\alpha < 1$ the fraction of ...
2
votes
2answers
333 views

Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture this object, and tell me something of its knot properties. Imagine vertical helices in $\mathbb{R}^3$, each ...
3
votes
2answers
249 views

Empty lattice simplex or White's theorem

White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem: If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of ...
9
votes
3answers
450 views

Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs of unit-radius (planar) circles, joined together at points where the tangents match. Thus the knot will have curvature $1$ at all but the ...
5
votes
2answers
244 views

Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$. Define a blocking set for a lattice cube to be a set of points in ...
3
votes
3answers
574 views

What fraction of a sphere's volume lies within a cone?

Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to ...
2
votes
2answers
494 views

Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood. I am considering a variant, which I call midpoint lattice polygons. Start with a sequence of ...
1
vote
0answers
285 views

Simple development of simple curve on a cone

Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting) curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve $\overline{C}$ on a plane by rolling ...
5
votes
2answers
571 views

Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including on a manifold (e.g., Melikyan, "Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"), but I have not seen this version: ...
9
votes
1answer
1k views

What nets fold to polyhedra?

There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
5
votes
0answers
287 views

Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected) pieces (where $k=1$ means just the square). Your task is to construct the largest volume convex body ...
11
votes
3answers
907 views

Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface of a cube by cutting up the square and pasting it onto the cube's surface. Q1. What is the largest ...
10
votes
3answers
678 views

Optimal Wireframe Sphere

Suppose you have a length $L$ of metal pipe at your disposal, and you would like to build a wireframe unit-radius sphere, by bending, cutting, and welding the pipe into a connected structure $F$. Your ...
1
vote
0answers
363 views

Uniform convergence of convex functions - references

Inspired by the following question on stackexchange: http://math.stackexchange.com/questions/126142/uniform-convergence-of-convex-sequence-of-functions, I thought of asking whether anyone knows of ...
10
votes
2answers
451 views

The “grassmannian” of a simplicial complex

This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called. Take an $n$ dimensional ...
10
votes
2answers
444 views

Periodic lightray paths trapped between two nested mirror circles

I wonder if the periodic paths of a lightray trapped between two nonconcentric circles, each perfectly reflecting, are known? The behavior of such rays seems chaotically complicated. For example, ...
2
votes
0answers
86 views

An equilibrium property of subsets of $\mathbb{R}^{n}$

Let $A\subset\mathbb{R}^{n}$ be a closed set and $\phi(x)=e^{-||x||^{2}}$, $x\in\mathbb{R}^{n}$. We shall say that $A$ is an "equilibrium set", if $x\in\partial A$ ...
17
votes
2answers
1k views

Probing a manifold with geodesics

Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$ embedded in $\mathbb{R}^3$. You do not know anything about $M$. You shoot off a geodesic $\gamma$ in some direction $u$, and learn ...
9
votes
3answers
718 views

Point sets in Euclidean space with a small number of distinct distances

It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
8
votes
1answer
532 views

Maximal tetrahedra inscribed in ellipsoid

Pietro Majer quoted the theorem of Michel Chasles in his MO question, "Convex curves with many inscribed triangles maximizing perimeter," which states that the triangles of maximum perimeter inscribed ...
22
votes
2answers
1k views

Why is the half-torus rigid?

The half-torus surface that results from slicing a torus like a bagel, depicted below (left), is isometrically rigid.       I know this from a remark of Alexandrov in Mathematics: ...
5
votes
1answer
305 views

Recovering a polyhedron from its tumble-density profile

Imagine a white convex polyhedron $P$ tumbling randomly about its fixed center of gravity (c.g.) $c$ against a blue background. A long-exposure photo would show pure white in a neighborhood of $c$ ...
3
votes
4answers
623 views

Relationship between the focal locus and the cut locus

I am seeking clarification of the relationship between the focal locus and the cut locus of a curve $C$ in $\mathbb{R}^2$, and of a surface $S$ in $\mathbb{R}^3$. Essentially my question is, Under ...
11
votes
1answer
641 views

On the definition of regularity

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules. (1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...
4
votes
5answers
453 views

Developable 3-manifolds in $\mathbb{R}^4$

Is there a classification of the equivalent of a "developable surface" in $\mathbb{R}^4$? Analogous to: planes, cylinders, cones, and tangent developables in $\mathbb{R}^3$? Edit: Here I am imagining ...
12
votes
2answers
680 views

Average degree of contact graph for balls in a box

Imagine you dump congruent, hard, frictionless balls in a box, letting gravity compress the balls into a stable configuration (I believe such configurations are called jammed.) Assume the box ...
5
votes
2answers
421 views

Shortest paths on linked tori

I will make this question specific at first, and general later. Suppose we have two linked tori, $T_1$ and $T_2$, each of radii $(2,1)$, meaning that each torus is the result of sweeping a circle of ...
7
votes
0answers
365 views

Who conjectured that a transitive projective plane is Desarguesian?

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...
0
votes
2answers
1k views

Parametrization of the intersection of an ellipsoid with a sphere

Hi, Fisrt I would like to say that geometry is far away from my domain. I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy ...
4
votes
0answers
105 views

What is known about the number of permissible simplicial complexes given the number of k-cells?

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
17
votes
3answers
2k views

Is there a combinatorial analogue of Ricci flow?

The question of generalising circle packing to three dimensions was asked in 65677. There is a clear consensus that there is no obvious three dimensional version of circle packing. However I have ...
13
votes
3answers
649 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
7
votes
0answers
226 views

Coloring toroidal polyhedra with convex faces?

Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 ...
7
votes
3answers
512 views

Shadow boundary on convex body in $\mathbb{R}^3$

Let $S$ be the surface of a compact, convex, smooth ($C^\infty$) body in $\mathbb{R}^3$, with strictly positive Gaussian curvature at every point of $S$. Fix a direction $z$ in a Cartesian coordinate ...
4
votes
1answer
335 views

analogues of Cayley plane as homogenous spaces

The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach ...