All Questions
4,828 questions
4
votes
1
answer
1k
views
Finding integer points on an N-d convex hull
Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is ...
- 151k
4
votes
3
answers
1k
views
Unit triangles with vertices on circles
Characterize all triples $c_1,c_2,c_3$ of circles in the plane such that
there are infinitely many unit regular triangles $a_1a_2a_3$ with $a_i\in c_i$ for $i=1,2,3$.
In particular, are there any ...
- 17.9k
8
votes
2
answers
621
views
Generalization of Hamiltonian cycles to "Hamiltonian spheres"
One possible generalization of a Hamiltonian cycle in a triangulated plane graph is what could be
called a Hamiltonian sphere: a collection of triangles within a simplicial complex in $\mathbb{R}^3$
...
- 24.7k
14
votes
7
answers
6k
views
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 ...
- 5,293
1
vote
0
answers
578
views
Cluster-preserving and distance-maximizing embedding into Hamming Space?
I have a set of data, each instance in the real $[0,1]^{d}$. However, it's actually all in a relatively small range around 0.5, clustered into classes in even smaller ranges. The actual origin of the ...
- 2,383
3
votes
1
answer
152
views
Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 44.3k
6
votes
0
answers
176
views
Spaces with the thin tetrahedra property
I read a comment about the $\delta$-thin tetrahedra property of a space.
It basically means, that if you choose any four points in this space, connect them by geodesics, and fill each triangle with a ...
- 11.1k
5
votes
1
answer
271
views
Feasibility of linear programs
It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
- 201
2
votes
2
answers
215
views
Is this a correct interpretation of support in coarse geometry?
Let $X = \mathbb{R}^n$, and consider a nondegenerate representation $\rho: C_0(X) \to B(H)$ where $B(H)$ is the algebra of bounded operators on a separable Hilbert space. The support of a vector $v \...
- 18.7k
2
votes
0
answers
5k
views
A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
- 13.5k
5
votes
1
answer
1k
views
Hyperbolic structure on surfaces with boundary
I have following two questions
1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). ...
- 14.4k
14
votes
1
answer
587
views
Pushing convex bodies together
Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies ...
- 1,618
0
votes
2
answers
176
views
Polygon Chain - Conversion to non-crossing while preserving shape?
I have polygon chains similar to the following...
http://upload.wikimedia.org/wikipedia/commons/thumb/6/62/Self_crossed_polygonal_chain.svg/220px-Self_crossed_polygonal_chain.svg.png
...given the ...
- 18.6k
4
votes
0
answers
790
views
Is it possible to use linear programming to solve this problem?
I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...
- 41
4
votes
2
answers
789
views
Is there any documented study of geometry in contemporary primates ? [closed]
There are many studies of language learning abilities of primates (mostly chimpanzee, bonobo) and studies of tool use, knowledge transmission and number sense.
Are there studies or documented cases ...
- 7,800
8
votes
1
answer
556
views
A variation on "Hearing the shape of a drum" for polytopes.
Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional ...
CommunityBot
- 1
10
votes
0
answers
1k
views
Dissecting trapezoids into triangles of equal area
[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
- 65.4k
1
vote
1
answer
419
views
Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 13.6k
4
votes
4
answers
7k
views
How to pick a random direction in n-dimensional space
I want to pick a random direction in n-dimensional space. How can I do this?
The reason I want to do this is to pick a neighbor for hill climbing optimization.
- 1,944
8
votes
1
answer
398
views
Möbius-invariant triangle center?
Given any two points x and y on a circle O, one can form four different lenses (regions between two circles, one of which is O) that have corners at x and y and make angles of 2π/3 at their corners. ...
- 85.6k
6
votes
3
answers
2k
views
A simple infinite dimensional optimization problem
I'd be grateful for a reference for the following result, which I believe to be true, and
should be well-known.
Let the continuous functions $f_0,f_1,\cdots,f_n: [0,1]\rightarrow [0,\infty)$ be ...
- 60.5k
5
votes
1
answer
1k
views
Do continuous maps give continuity in the 'topology' of Hausdorff distance?
I was reading this question:
limiting behaviour of converging loops on a torus
And I wanted to be able to give an argument along the lines of: "If your loops are converging in your torus, their ...
CommunityBot
- 1
4
votes
2
answers
271
views
Centralizing four red vectors in six green sectors
Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid star of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
\...
- 18.6k
1
vote
2
answers
1k
views
Calculating the surface area distribution of two-dimensional projections for a polytope
My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex ...
CommunityBot
- 1
6
votes
3
answers
982
views
Boolean network as a gauge field
Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two ...
- 2,179
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
9
votes
3
answers
1k
views
What are trig classes like within a universe that's "noticeably" hyperbolic?
[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]
What are trig classes like within a universe that's "noticeably"[*] ...
- 1,723
7
votes
2
answers
846
views
What is known about polyhedra nets that allow overlapping?
It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
5
votes
1
answer
491
views
Isometric embedding of a positively curved polyhedral surface
Suppose you have a 2-dimensional polyhedral surface with specified lengths for the edges so that the vertices all have positive curvature. I believe this has a unique isometric embedding into 3-...
CommunityBot
- 1
4
votes
3
answers
1k
views
Topological embeddings of non-compact, complete metric spaces
Given a completely metrizable space, say that it has property X if it can be embedded in some metric space such that its image is not closed. For example, the real line R can be embedded, ...
- 4,060
5
votes
2
answers
523
views
Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.
Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries?
The answer is easy for $k=1$....
- 6,523
4
votes
1
answer
246
views
Name for an inequality of isoperimetric type
I want to know if the following fact has a standard name and/or reference
Let $X$ be a subset of $\mathbb R^2$ and $B$ be a disc of the same area as $X$.
Set $X_\epsilon$ to be the $\epsilon$-...
- 85.6k
5
votes
2
answers
835
views
Diameter of a circle in an embedded Riemannian manifold
This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of ...
CommunityBot
- 1
8
votes
0
answers
588
views
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 ...
- 3
4
votes
2
answers
394
views
Is a "contraction space" always complete?
Some of the fundamental results in analysis (inverse function theorem, existence and uniqueness of solutions to ODEs) have slick proofs using the idea of a contraction. So, it seems plausible to me ...
- 4,060
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
4
votes
1
answer
210
views
Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets
Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha_2(A)))-\log(\mathrm{Vol}(\alpha_1(A)))$ where $\alpha_1,\alpha_2$ are two affine bijections such ...
- 11.4k
3
votes
1
answer
147
views
Optimizing finite-length approximations to space-filling loops
Take a loop in the unit disk D^2, with length l where length is defined as the supremum of the lengths of piecewise linear approximations. What is the smallest r such that every radius-r subdisk of D^...
- 14.4k
13
votes
2
answers
917
views
Can the circle be characterized by the following property?
In the Euclidean plane, is the circle the only simple closed curve that has an axis of symmetry in every
direction?
- 7,800
10
votes
1
answer
2k
views
Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 61.9k
11
votes
1
answer
1k
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 ...
- 45k
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
6
votes
1
answer
589
views
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 ...
- 4,577
7
votes
4
answers
3k
views
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 ...
- 758
2
votes
1
answer
247
views
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 ...
CommunityBot
- 1
3
votes
2
answers
1k
views
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 &...
- 879
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 ...
- 45k
5
votes
1
answer
586
views
a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
- 10.8k
14
votes
4
answers
2k
views
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 ...
CommunityBot
- 1
3
votes
1
answer
325
views
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 ...
- 45k