Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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22
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Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
51
votes
3answers
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Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth ...
15
votes
9answers
7k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
13
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8answers
2k views

Representability of finite metric spaces

There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces. Suppose $X$ is a set equipped with a metric ...
26
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0answers
914 views

Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ...
4
votes
1answer
203 views

Computational approach deciding whether a set of Wang Tile could tile the space up to some size

As an applied person, I'm facing one practical problem deciding whether a set of Wang tile could tile the plane periodically or aperiodically. Although both problems seem undecidable, but I'm on a ...
53
votes
9answers
5k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
39
votes
7answers
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Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
39
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12answers
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Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximatively equal to the area of the disc. Does the complement of S necessarily contain ...
45
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5answers
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If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds. Monsky's famous and amazingly tricky proof says that if we ...
33
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1answer
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Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown. I have already asked this question on ...
11
votes
3answers
427 views

Can a tangle of arcs interlock?

Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining ...
4
votes
5answers
903 views

Feasibility of a list of prescribed distances in R^3

I am puzzled with the following problem: Given $n$ real numbers it is to obtain a Yes/No answer to: "whether it is possible to arrange different points in the Euclidean $\mathbb{R}^3$ so that every ...
19
votes
3answers
763 views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
7
votes
2answers
358 views

Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set $$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$ Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by ...
11
votes
3answers
744 views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. ...
9
votes
1answer
289 views

Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction? The answer does ...
8
votes
1answer
628 views

In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?

A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...
4
votes
4answers
928 views

Delaunay triangulations and convex hulls

This is a reference request. I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
12
votes
1answer
343 views

(A question about)${}^3$ 3-dimensional convex bodies

Related to the questions mathoverflow.net question No. 137850 and mathoverflow.net question No. 39127, is there a 3-dimensional convex body other than a ball whose perpendicular projections in all ...
5
votes
1answer
661 views

A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...
6
votes
2answers
469 views

Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying : $X_{n}$ have topological dimension $n$. $X_{n+1}$ is ...
6
votes
1answer
228 views

A question about a question about 3-dimensional convex bodies

For each positive integer n let E(n) denote n-dimensional Euclidean space and let the term "n-dimensional convex body" mean a compact convex subset of E(n) whose interior (with respect to E(n)) is ...
5
votes
1answer
126 views

Maximal regions with given diameter

Let us call a bounded region $D$ in the plane maximal if the conditions $D\subset D'$ and $\mathrm{diam} D'=\mathrm{diam} D$ imply $D'=D$. Is it possible to describe all maximal regions? The only ...
4
votes
3answers
530 views

Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below : This post has been divided into two parts, the second part is here. Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
4
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0answers
155 views

Upper bounds on art gallery problems using independent witnesses

Given a polygon $P$, the art gallery problem looks to find a smallest set of points that sees all of $P$. One way of bounding the number of guards necessary (from below) is to find a largest set of ...
4
votes
0answers
223 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 $$ ...
21
votes
6answers
2k views

Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of radius ...
52
votes
6answers
4k views

Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
62
votes
11answers
8k views

Is it possible to capture a sphere in a knot?

You and I decide to play a game: To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
15
votes
1answer
334 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
15
votes
5answers
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Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...
33
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3answers
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$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$

I read (in a paper by Emil Saucan) that the flat torus may be isometrically embedded in $\mathbb{R}^3$ with a $C^1$ map by the Kuiper extension of the Nash Embedding Theorem, a claim repeated in this ...
34
votes
3answers
1k views

What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
25
votes
3answers
950 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
20
votes
1answer
468 views

Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$, the function $p_V\colon \mathbb R_+\to \mathbb R_+$ $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$. Definition 2. Define ...
19
votes
5answers
806 views

Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given point $p$ such that it gets "stuck" reflecting between two congruent mirror-disks. For why there is such a ray, see the (amazing!) answer ...
17
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9answers
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Open problems in Euclidean geometry ?

Which are some (research level) open problems in Euclidean geometry ? (Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet) I should clarify a ...
7
votes
0answers
101 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
27
votes
2answers
731 views

Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
22
votes
4answers
1k views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
12
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4answers
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What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
8
votes
3answers
927 views

Intuition for Levi-Civita connection via Hamiltonian flows

A Euclidean metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?
17
votes
2answers
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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 ...
16
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2answers
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Is there a neat formula for the volume of a tetrahedron on the surface of $S^3$?

There is a nice formula for the area of a triangle on the surface of the 2-dimensional sphere; If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, ...
15
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5answers
893 views

Is a rhombus rigid on a sphere or torus? And generalizations.

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a ...
21
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2answers
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Non-regular Connected Hausdorff Banach Manifold

After reading this MO post, I am wondering: Is every (connected) Hausdorff Banach manifold a regular space? Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
17
votes
2answers
467 views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
16
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5answers
2k views

Advanced view of the napkin ring problem?

The "napkin-ring problem" sometimes shows up in 2nd-year calculus courses, but it can fit quite neatly into a high-school geometry course via Cavalieri's principle. However, the conclusion remains ...
14
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1answer
1k views

Hanging a ball with string

What is the shortest length of string that suffices to hang a unit-radius ball $B$? This question is related to an earlier MO question, but I think different. Assume that the ball is ...