All Questions
4,827 questions
2
votes
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237
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Geometric interpretation of trace of a linear operator
This question is really an addendum to Geometric interpretation of trace
There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
- 118k
1
vote
0
answers
114
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A circle is inscribed in a triangle, with three other circles in the corner regions. The radii are integers. Possible values of the largest radius?
Originally posted at MSE.
A circle with integer radius $R$ is inscribed in a triangle. Three other circles with integer radii $a,b,c$ are each tangent to the large circle and two sides of the ...
- 3,527
4
votes
2
answers
299
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Is there a set of point $S \subset \mathbb R^2$ such that $|\{C: C \text{ is unit circle boundary }, |C \cap S| = 10\}| > |S|$
There are some blue points and red points on the plane such that in the boundary of every unit circle centered at one blue point there are exactly 10 red point. Can the number of blue points strictly ...
6
votes
2
answers
207
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Volume satisfying inequality constraints (simplex subset)
Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...
- 111
124
votes
37
answers
12k
views
One-step problems in geometry
I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you ...
- 2,152
0
votes
0
answers
82
views
On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
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3
votes
0
answers
208
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Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
3
votes
2
answers
279
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Construct by compactness (Pentagonal tiling – Rao paper)
In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:
"… We keep a connected component $H_d'$ of $...
- 1,341
9
votes
2
answers
658
views
Probability that randomly chosen balls have a nonempty common intersection
Fix some $0 < r < 1$. A collection of points $x_1, \dots, x_n$ are chosen independently and uniformly at random from the closed unit ball in $\mathbb R^d$.
What is the probability that the ...
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7
votes
1
answer
497
views
Is there a bicyclic irregular pentagon in integers?
Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well?
If it does ...
- 2,370
12
votes
1
answer
373
views
A claim on partitioning a convex planar region into congruent pieces
Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
- 4,713
8
votes
1
answer
4k
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Covering number of Lipschitz functions
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
That the $\infty$-...
- 1
4
votes
2
answers
1k
views
Polyline averaging
I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...
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6
votes
3
answers
2k
views
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 ...
- 4,219
9
votes
1
answer
281
views
Is every compact smooth Riemannian manifold bilipschitz equivalent to a finite simplicial complex?
Let $M$ be a compact smooth Riemannian manifold. Then it admits a triangulation, i.e. a finite simplicial complex $K$ which is homeomorphic to $M$. Any such simplicial complex carries a natural metric ...
- 45k
0
votes
0
answers
173
views
A question on Cheeger-Colding theory
I'm reading Compactification of certain Kähler manifolds with nonnegative Ricci curvature by Gang Liu recently. And I feel hard to understand a statement in the paper. Now the assumption is $(M,g)$ is ...
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1
vote
1
answer
134
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An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
- 5,979
8
votes
1
answer
567
views
Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space).
We need to ...
- 156k
3
votes
1
answer
427
views
Generalization of some plane geometry theorems
Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\...
- 109k
7
votes
1
answer
246
views
Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
3
votes
0
answers
136
views
If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
- 5,979
15
votes
2
answers
1k
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Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B ...
- 225
3
votes
1
answer
253
views
Nagel line of a tetrahedron?
It's well known that there is an analogy for the Euler line in a tetrahedron, but is there also an analogy for the nagel line of a tetrahedron? I can't seem to find any decent literature talking about ...
- 109k
4
votes
1
answer
360
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 ...
2
votes
1
answer
196
views
Cusp points in Alexandrov spaces
Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
CommunityBot
- 1
0
votes
1
answer
604
views
Finding lattice with short basis-vectors containing given lattice
While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$, I am would like to find ...
- 63.9k
0
votes
0
answers
89
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Practical applications of dandelin spheres
I know that dandelin spheres can be used to prove the focal properties of conic sections, but I heard that they can be used to help track the orbits of planets. All the sources I looked up only said ...
- 12.9k
1
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0
answers
42
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On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
0
votes
0
answers
115
views
Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
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1
vote
0
answers
59
views
What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
2
votes
1
answer
302
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Are two convex solids with all corresponding shadows equal in area congruent?
By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
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94
votes
2
answers
6k
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Volumes of sets of constant width in high dimensions
Background
The $n$-dimensional Euclidean ball of radius $1/2$ has width $1$ in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between ...
- 24.7k
4
votes
1
answer
143
views
On the history of cone-3-manifolds
A cone-3-manifold (of constant curvature) is a geometric 3-manifold locally modelled either on the Euclidean/hyperbolic/spherical 3-space or on the respective metric cones over spherical cone-surfaces ...
- 28.1k
5
votes
1
answer
176
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Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 5,429
1
vote
1
answer
255
views
How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
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1
vote
0
answers
112
views
Is the formula known? and can we generalized for higher dimensions?
In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?
- 4,713
2
votes
1
answer
600
views
A geometric approach to the odd perfect number problem?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
- 3,054
1
vote
0
answers
72
views
Special rectangle and its existence in non-Euclidean geometries
My questions is motivated by Folding the Hyperbolic Crane article which presents non-Euclidean paper for origami and the existence of a special rectangle on Euclidean paper.
Actually, there exists a ...
- 5,429
30
votes
0
answers
1k
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Curves on potatoes
On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler:
The puzzle is attributed to the book "The mathemagician and pied puzzler", and ...
- 2,896
6
votes
2
answers
404
views
Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
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19
votes
3
answers
2k
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Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
- 646
11
votes
0
answers
734
views
Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is false (...
- 4,713
4
votes
1
answer
303
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On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
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12
votes
1
answer
380
views
Do lattices with small covering radius have sublattices with small covering radius?
For me a lattice is a discrete subgroup of $\mathbb R^n$. The linear span of a lattice, written $\Lambda \otimes \mathbb R$, is the $\mathbb R$-vector subspace of $\mathbb R^n$ generated by $\Lambda$. ...
- 5,382
19
votes
3
answers
2k
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Towards a metric characterization of Euclidean spaces
I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry.
Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each ...
- 45k
4
votes
2
answers
154
views
Pushing a convex cone and equidistants
Let $K$ be a closed convex cone in an n-dimensional Euclidean space.
Suppose $K$ has non-empty interior. For $t > 0$
form the subcone $K_t$ consisting of all points in $K$ which lie a ...
- 8,336
1
vote
2
answers
164
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General and translational Birkhoff lattices. Equational classes
By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
Is there an equational class ...
- 14.6k
0
votes
0
answers
47
views
Rauch comparison theorem for $C^{1,1}$ metrics
If $g$ is a smooth riemannian metric on $M$ with nonpositive sectional curvature, the Rauch comparison theorem implies that $(M,d_g)$ is a negatively curved metric space (every point has a ...
- 151
1
vote
0
answers
42
views
Genaralizing the metric expression present in the quadrilateral inequality
Let $(X, d)$ be a metric space. In Sato - An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality it is stated that if $X$ is a geodesic space, then ...
- 12.9k
1
vote
0
answers
56
views
Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
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