All Questions
4,827 questions
11
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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 ...
- 32.4k
14
votes
7
answers
6k
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The Symmetry of a Soccer Ball
Let $P$ be a polyhedron which satisfies the following three conditions:
$P$ is built out of regular hexagons and regular pentagons.
Three faces meet at each vertex.
$P$ is topologically a sphere.
An ...
- 865
3
votes
5
answers
1k
views
Approximate solutions for trisecting the angle or squaring the circle
Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a ...
- 3,595
20
votes
3
answers
3k
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How many unit squares can you pack into a rectangle with nearly integer side lengths?
Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting:
Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ is ...
102
votes
6
answers
11k
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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 ...
- 29.2k
5
votes
1
answer
378
views
How far can the analogy between a Cayley graph and a symmetric space be pushed?
If $G$ is a finitely group and $S$ a finite symmetric set of generators, the associated Cayley graph, then $x \mapsto x^{-1}$ gives rise to a geodesic symmetry $i$ at the identity:
If $g=s_1^{e_1}\...
- 4,280
94
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5
answers
9k
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Is there a dense subset of the real plane with all pairwise distances rational?
I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
- 236k
6
votes
1
answer
589
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Generalizing cosine rule to symmetric spaces
The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
user2529
8
votes
1
answer
1k
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Did Apollonius invent co-ordinate geometry?
When I read descriptions of Apollonius' treatise on conics, some of them say that he invented co-ordinate geometry, some say that he kind of did and others are silent on the matter. Or is it the case ...
- 4,351
2
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1
answer
247
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Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?
This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
6
votes
2
answers
433
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Triangles, squares, and discontinuous complex functions
Is there some onto function $f:$ $\mathbb{C}$ $\rightarrow$ $\mathbb{C}$
such that for each triangle $T$ (with its interior), $f(T)$ is a
square (with interior, too) ?
I would have the same question ...
- 63
7
votes
4
answers
3k
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Existence of Fermi coordinates on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
- 8,512
16
votes
2
answers
3k
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Metric on one-point compactification
Is there a standard construction of a metric on one-point compactification of a proper metric space?
Comments:
A metric space is proper if all bounded closed sets are compact.
Standard means found in ...
- 45k
4
votes
2
answers
575
views
Routh's theorem in three dimensions
The most well known case of Routh's triangle theorem is:
If the sides BC, CA,and AB are trisected at the points D, E, and F, respectively, then the area of the inside triangle formed by AD, BE, ...
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62
votes
7
answers
26k
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Is the Jaccard distance a distance?
Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...
- 1,355
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 32.4k
8
votes
1
answer
447
views
Stable Tables on Fluctuating Floors
If a four-legged, rectangular table is rickety, it can nearly always be stabilised just by turning it a little. This is very useful in everyday life! Of course it relies on the floor being the source ...
- 2,251
3
votes
2
answers
1k
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Are Bregman divergences quasi-convex?
Given a convex set S ⊂ ℝd and an appropriately differentiable convex function f: S → ℝ, a Bregman divergence Bf(x, y) = f (x) - f (y) -⟨x- y , ∇f (y)⟩ for x, y &...
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11
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1
answer
1k
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54
votes
3
answers
11k
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Sheaves and bundles in differential geometry
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
- 19.6k
1
vote
1
answer
524
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How to compute the number of regular spheres needed to fill a rectangular space
Computing the volume of a sphere is straightforward 4/3*pi*R^3
As is the volume of a rectangular space length*width*height (e.g. 10*10*6)
How might I go about determining how many spheres would fit ...
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7
votes
4
answers
3k
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completeness axiom for the real numbers
Do any treatises on real analysis take the following as the basic completeness axiom for the reals?
"Let $A$ and $B$ be set of real numbers such that
(a) every real number is either in $A$ or in $B$;
...
- 19.7k
14
votes
4
answers
2k
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When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?
Suppose we have a surface S (although the question might make as much sense in higher dimensions) and a topological group G. The data of a flat vector bundle on S (up to isomorphism) is the same as a ...
- 3,203
2
votes
4
answers
8k
views
Compute the Centroid of a 3D Planar Polygon
Given a list of 3D coordinates that define the surface( Point3D1, Point3D2, Point3D3, and so ...
- 381
8
votes
3
answers
540
views
Set of vectors separated by at least a specified angle
Suppose $\theta$ and $d$ are given.
How big can a set of $d$-dimensional vectors be such that no pair of them are at angle less than theta?
I particularly want an upper bound; that is, an $n=n(\...
5
votes
2
answers
484
views
Better term for a (simplicial) contractible plane continuum
In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also ...
- 56.6k
3
votes
1
answer
325
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Is Level set of Regular functions in Alexandrov spaces again an Alex. space?
Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space?
In my mind, the intrinsic metric on the level set is 'comparable' to ...
- 1,101
12
votes
2
answers
1k
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Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses.
Let E be an ellipse centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the y-...
- 2,374
2
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2
answers
3k
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Is ellipse on a sphere convex? (proof)
Is 'small enough' ellipse projected on a surface of a sphere convex? By ellipse I mean a set of points 'C' with a constant sum |AC| + |BC|, A and B are the centers. By 'small enough' I mean that the ...
- 23
5
votes
3
answers
1k
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Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?
Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
26
votes
7
answers
2k
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Tetrahedra with prescribed face angles
I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
- 11.1k
4
votes
1
answer
207
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Rectifying texture from image
I have a camera matrix $P$ which defines a projective transformation $\mathbb{P}^3 \rightarrow \mathbb{P}^2$. In the former space there is a plane $[ x|\pi^Tx=0 ]$. The image of the plane under $P$ ...
- 567
6
votes
3
answers
2k
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Constructing a metric over a lattice
Consider a lattice $({\cal L}, \wedge, \vee)$ with an antimonotonic function $f: {\cal L} \rightarrow {\mathbb R}$ defined on it (i.e $x \preceq y \implies f(x) \ge f(y)$).
$f$ is said to be ...
- 4,462
16
votes
4
answers
2k
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Inverse limit in metric geometry
Question. Did you ever see inverse limits to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?
The definition of inverse limit for metric spaces is given ...
4
votes
2
answers
2k
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Are isometries the only geodesic preserving maps in a CAT(0)-space?
Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with ...
- 11.1k
4
votes
2
answers
863
views
Soul theorem for non-negativly curved open Alexandrov manifolds?
Alexandrov manifold means Alexandrov space which happens to be a manifold, i.e. the space of directions is homeomorphic to shpere. Sorry for introducing this new term.
For such a open manifold does ...
- 1,101
23
votes
4
answers
3k
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Intrinsic metric with no geodesics
It seems that I have the needed example, but I want it to be simple and self-explaining...
Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing ...
- 45k
6
votes
2
answers
2k
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Problem equivalent to "largest square in a cube"
The "largest square in a cube" problem, which asks for the largest square inside a cube, has a solution as can be seen on this page, which also says that the general problem in higher dimensions is ...
- 7,320
2
votes
0
answers
254
views
Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
6
votes
2
answers
631
views
How to define a Voronoi reduced basis?
Let $\Lambda$ be an $n$-dimensional lattice with basis $b_1,\ldots,b_n$. The problem of finding a "good" basis for $\Lambda$, or reducing a "bad" basis into a good one, is a very active area of ...
- 1,085
28
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3
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Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
- 18.7k
19
votes
5
answers
21k
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Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
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1
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Can the algebraic closure of a complete field be complete and of infinite degree?
Yes, this is yet another "foundational" question in valuation theory.
Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real ...
- 65.4k
20
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5
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3k
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Finding Constant Curvature Metrics on Surfaces without full power of Uniformization
(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)
Let S be a surface (possibly non-compact, but no boundary). It seems that there are ...
- 3,203
3
votes
1
answer
1k
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Maximizing Sparsity in l1 Minimization?
Consider the optimization problem
$$\min_x ||Ax||_1 + \lambda||x-b||^2,$$
where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is ...
- 3,059
3
votes
1
answer
2k
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Finding a minimum bounding sphere for a frustum
I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...
- 31
10
votes
1
answer
640
views
Ordered geometries from convex subsets of the plane
Motivation
In the Klein disk model of the hyperbolic plane, the points are the interior of the disk, and the lines in $H^2$ correspond to lines intersecting the interior.
Similarly, the Euclidean ...
- 28k
5
votes
4
answers
1k
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Coloring Points in the Plane
Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...
- 5,275
8
votes
0
answers
588
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Hausdorff measure question
Say we have some compact metrisable topological space $X$ with a measure $\mu$ defined on the Borel sets of $X$. Then is there some way to determine whether $\mu$ is the Hausdorff measure associated ...
- 1,665
5
votes
1
answer
782
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Malfatti Circles - Limiting point
"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see ...
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