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

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1answer
78 views

general formula for volume of a simplex?

I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to ...
7
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1answer
199 views

The Minkowski sum of two curves

Let $\gamma$ be a continuous curve in the complex plane without self-intersections and let $\lambda$ be a complex non-real number less than 1 in modulus. Put $\gamma'=\lambda\gamma$. Question. Is it ...
7
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0answers
61 views

Approximate singular value decomposition in Banach spaces

I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are ...
10
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1answer
241 views

Soft question: mathematics about truchet tiles

It seems that this is the first question on Truchet tiles on MO. Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells. I ...
3
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1answer
149 views

n-dimensional Delaunay Triangulation of Lattices

I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
-3
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0answers
28 views

Determine angle of a straight line in 3D space [on hold]

I have a straight line in space with an start and end point (x,y,z) and I am attempting to get the angle but I cannot figure out the formula for this. I am using VB.NET Here is a picture of the line ...
1
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1answer
200 views

Does the singular cohomology of a metric space of finite topological dimension vanish in high dimensions?

It is known that by applying the universal coefficient theorem, the singular cohomology of closed manifold with coefficient $\mathbb{Z}_2$ vanishes in high dimensions. But for a metric space $M$ with ...
2
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1answer
75 views

Parallel transport on a Hadamard manifold

Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and ...
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1answer
61 views

Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference? Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected subcomplex of $X$. Then the following are equivalent: ...
3
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1answer
63 views

conjugacy between geodesic flows on 2-tori

Let $(T_1,g_1)$ and $(T_2,g_2)$ be two flat tori of dimension 2 such that their geodesic flows are $C^0$-conjugated, is there an isometry between $(T_1,g_1)$ and $(T_2,g_2)$ ? I emphasize the fact ...
6
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1answer
126 views

Embedding Euclidean buildings into products of trees

A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.) Question: Is it true ...
7
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2answers
344 views

Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...
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0answers
30 views

Covering number of the range of a function

I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
18
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2answers
682 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 ...
10
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1answer
411 views

Metric $d(A,B) = \mathbb P(\overline A\cup\overline B\mid A\cup B)$

I'm wondering where the relative probabilistic distance was first studied: $$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$ where $\overline A$ is the complement of $A$. A web search ...
6
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1answer
244 views

Which surfaces have only a finite number of connecting geodesics?

Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$, under which conditions is it true that, for every pair of points $a,b \in S$, there are an infinite number of ...
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3answers
491 views

In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line. In which other geometries does this hold?
4
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1answer
126 views

Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
18
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5answers
2k views

Unexpected applications of Dvoretzky's theorem

Dvoretzky's theorem is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem ...
3
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0answers
70 views

Surfaces with curvature $\leq K$ are of bounded integral curvature

One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$ is the ...
4
votes
2answers
309 views

Distance function to a submanifold

Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where ...
1
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1answer
148 views

Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$ I came across the following generalization of this equality: $$\sqrt{1-k^2} {\rm ...
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0answers
38 views

Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)

I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...
8
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1answer
105 views

Tilting the $d$-cube to vertically separate its vertices

Let $C_d$ be a unit edge-length cube in $d$ dimensions. I would like to orient it ("tilt" it) so that the vertical (last) coordinates of its $2^d$ vertices are maximally separated, in the sense that ...
2
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1answer
188 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
6
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3answers
407 views

Polynomial threading through a monotone corridor

I have a need to find a polynomial of minimal degree that connects two points and stays within a given "corridor," by which I mean an $x$-monotone polygon. Here is an example:       ...
12
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1answer
341 views

Lower-Hölder embeddings of the sphere

My question is very simple: Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that $$ |f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r, $$ for ...
4
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1answer
463 views

Focus of parabola using only a ruler

It is an easy exercise that using ruler and compass one find the focus of a given parabola. Can one do the same using only a ruler? -- if not, why?
3
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1answer
275 views

The Praying Eyes theorem generalized

Is there an obvious way to slice two spheres (no necessarily equal) simultaneously such that the sections share common areas? I mean, I cannot see another way (easier) different from the way provided ...
1
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0answers
91 views

Equidissection of square [duplicate]

Monsky's Theorem states: One cannot dissect a square into an uneven number of triangles with equal area. I was wondering how close one could get to a equidissection, i.e. For $n$ an uneven ...
3
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1answer
148 views

The relation between Hausdorff dimension of an $n$-manifold and $n$

It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general. For the case of manifolds, suppose $M$ is a $n$-manifold with a ...
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0answers
103 views

Shortest rope to capture a sphere of diameter 1 [duplicate]

I have a perfect rigid sphere of diameter 1. I have infinite supply of rope. The rope is infinitely flexible and can be cut or glued without losing or adding length. The rope can be glued at any ...
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0answers
42 views

Length invariance under nondecreasing changes of parameters

Suppose that $f\colon [0,1]\to [0,1]$ is a continuous, surjective and nondecreasing function, for example the Cantor function. Let $X$ be a metric space (not necessarily a length space) and let $L$ be ...
8
votes
5answers
2k views

Subset of the plane that intersects every line exactly twice

In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to ...
24
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1answer
593 views

The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
0
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0answers
20 views

How to calculate the center of a regular polygon? [migrated]

What is the formula for the center of an n-edge regular polygon that has the given segment as its edge? So, given a segment AB, ...
11
votes
1answer
280 views

Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties: $M(a,a)=a\qquad$ (identity) $M(a,b)=M(b,a)\qquad$ (commutativity). and possibly ...
1
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2answers
131 views

Regular paths along surface of sphere

I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$. The movement should be repetitive, so that ...
4
votes
1answer
209 views

convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where ...
3
votes
2answers
276 views

Lipschitz parametrization of a symmetric convex curve

Assume that $\gamma$ is a convex curve which is symmetric with respect to two orthogonal lines (the ellipse is such a curve). I want to know if there exists a $(l,L)$-bi-Lipschitz mapping of the ...
36
votes
3answers
2k views

Is the “Napkin conjecture” open? (origami)

The falsity of the following conjecture would be a nice counter-intuitive fact. Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
8
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2answers
1k views

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? That is, given a line and a point not on the line, construct a line parallel to the given line.
13
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2answers
959 views

Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection). I am wondering if each of the five ...
27
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3answers
1k views

Diameter of m-fold cover

Let $M$ be a closed Riemannian manifold. Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$. Is it true that $$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (*)$$ ...
5
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2answers
341 views

Generalization of Pascal's Theorem to Higher Dimensions

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
6
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0answers
75 views

Unbalanced equipartitions

Let $K$ be a compact convex set in the plane. Say that a perimeter-halving partition of $K$ is a partition of $K$ into two pieces by a chord (a segment with endpoints on the boundary $\partial K$) ...
42
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2answers
2k views

The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose internal face surfaces were perfect mirrors. Let's assume $T$'s height is $3{\times}$ yours, so that your eye is roughly at the ...
1
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1answer
67 views

Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$ consider the (energy) functional ...
6
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3answers
188 views

Most dispersed set of points in a disk?

Put $1$ billion points in a disk of radius $1$. Consider the minimal area $A$ of a triangle formed by any $3$ points. Where do you put the points so that $A$ is maximal and how much is $A$? Consider ...