All Questions
4,827 questions
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 949
21
votes
5
answers
1k
views
Is there a midsphere theorem for 4-polytopes?
The (remarkable) midsphere theorem says that each combinatorial
type of convex polyhedron may be realized by one all of whose edges are
tangent to a sphere
(and the realization is unique if the center ...
- 151k
4
votes
1
answer
631
views
Reformulation of amenability and growth rate of a group in terms of general metric spaces.
Updates: Changed a bit the definition to include infinite dimensional Banach spaces; Included questions 0 and subquestion. ...
- 6,034
22
votes
1
answer
663
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 ...
- 45k
3
votes
1
answer
316
views
Hausdorff dimension and Mertens function
Hello,
when one plots the Mertens function, it really looks like a fractal. So does anyone know the (approximate) value of the Hausdorff dimension of the set $\{(x,y),y=M(x),x\in\mathbb{R}^+\}$?
...
- 6,984
16
votes
1
answer
1k
views
Random polycube shapes
I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
(...
- 151k
7
votes
0
answers
669
views
Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
- 151k
2
votes
5
answers
6k
views
Quadrilateral from 4 random points
Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...
- 123
23
votes
3
answers
2k
views
Rolling-ball game
The analyses
in two recent MO questions
("recent" with respect to the original posting in 2011),
"Rolling a random walk on a sphere"
and
"Maneuvering with limited moves on $S^2$,"
suggest a Rolling-...
- 151k
3
votes
1
answer
281
views
Maximum distance of points in intersection of balls
let $B_\delta(p):=\{x\in\mathbb{R}^d:||x-p||_2\leq \delta\}$ be a $d$-dimensional closed ball.
Now I do not have one ball, but four:
$B_{r_1}(p)$, $B_{r_2}(p)$, $B_{s_1}(q)$ and $B_{s_2}(q)$. ...
- 255
1
vote
1
answer
234
views
Model for shipping widgets in an optimal way
I am a programmer and have the following requirement.
We are trying to figure out the optimal way to ship widgets. Below is the scenario:
We need to ship 1,000,000 widgets
We have two different size ...
- 113
22
votes
5
answers
2k
views
Which norms have rich isometry groups?
Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections $F:\...
- 4,874
5
votes
2
answers
657
views
Area of intersection of a family of circles in the plane
Suppose you are given a family F of circles in the plane such that each circle has radius 1. Let G be the family of circles with same centers as in family F but now each circle has radius $r$. Let A ...
- 51
1
vote
0
answers
785
views
A curious property of the Gergonne point
Ha, finally no knot theory :-)
First of all, let's define the "power line" of three circles.
(Very probably, someone had the idea before me, but no math forum
ever came up with something .)
Call the ...
- 4,829
16
votes
1
answer
806
views
Blocking visibility with cylinders
Suppose you have a supply of infinite-length, opaque, unit-radius cylinders,
and you would like to block all visibility from a point
$p \in \mathbb{R}^3$ to infinity with as few cylinders as possible.
...
19
votes
3
answers
2k
views
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 ...
- 1,987
50
votes
4
answers
6k
views
The maximum of a polynomial on the unit circle
Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...
- 23k
10
votes
1
answer
688
views
$G$-structures of finite type.
A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie ...
- 355
11
votes
4
answers
4k
views
Eigenvalues of Laplacian-Beltrami operator
I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value.
I am interested in the case when ...
- 163
4
votes
2
answers
1k
views
Set Cover:Greedy vs LP
Hi
Both, the greedy and the LP approach for Set Cover give a O(log n) approximation. Is there some inherent difference on the two approximation approaches?
thanks
- 239
4
votes
3
answers
2k
views
Minimum norm of convex hull
I am currently stuck at a problem which seems too easy to be stuck at to me...
Summary
Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute
\[\min_{x\in H}...
- 255
-1
votes
1
answer
467
views
Meeting point of the vertices of a square cloth on x-y plane [closed]
Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...
- 101
45
votes
1
answer
4k
views
Rolling a random walk on a sphere
A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...
- 151k
3
votes
5
answers
813
views
Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
- 309
3
votes
3
answers
3k
views
How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?
I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot).
Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...
- 133
2
votes
0
answers
147
views
System dynamic of space euclidean and hyperbolic tilings
Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC)
tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of
$R^{d}$ by translation is on ...
- 21
13
votes
3
answers
1k
views
Efficient visibility blockers in Pólya's orchard problem
Pólya'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.
It has been ...
- 151k
2
votes
1
answer
304
views
existence of l1 embedding using LP feasibility
hello
Let (A, d) be an n-point metric space
for $t \geq 1$,the task it to find an integer $m$ and an embedding $f : A \rightarrow R^m$ s.t.
$\forall x,y \in A$ : $d(x,y) \leq d_1(f(x), f(y)) \leq t*...
- 239
1
vote
1
answer
531
views
Split sum into equal terms
Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$.
Find indices
$1 < p_1 <...< p_h <...< p_{t-1} < l$
such that in sum
$(r_1+...+r_{p_1})+...+(r_{p_{h-1}+1}+......
- 11
30
votes
2
answers
2k
views
Maneuvering with limited moves on $S^2$
This question comes to me via a friend, and apparently has something to do with quantum physics. However, stripped of all physics, it seems interesting enough on its own. I assume someone has asked ...
- 1,318
26
votes
2
answers
13k
views
Geometric interpretation of Cartan's structure equations
Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by $\Omega_i^...
- 2,035
2
votes
4
answers
2k
views
Efficient algorithm for finding the minima of a piecewise linear function
Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two lines:...
- 823
0
votes
1
answer
2k
views
Quaternion between two quaternions [closed]
Hello,
I have an orientation P1 in a 3D space, represented as a quaternion [w x y z].
Then P1 is rotated using another quaternion (q1) with the formula
P2=q1*P1*q1'...
- 9
5
votes
2
answers
1k
views
Critical Radius for Infinite Dimensional Sphere Packing
Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.
For example, when $r=1/(1+\sqrt2)$...
- 543
2
votes
1
answer
404
views
Alexandrov curvature of a compact length space
I've found lots of (more or less precise) definitions of the Alexandrov curvature, but I'm mainly interested in that of "Alexandrov curvature bounded below". Could anyone give me that or give me a ...
- 6,034
21
votes
1
answer
1k
views
Which convex bodies roll along closed geodesics?
An ellipsoid could be rolled (without slippage) on a horizontal plane so that its point
of contact traces out a closed geodesic on its surface:
...
- 151k
0
votes
2
answers
333
views
Explicit example of a smooth - but not analytic- closed curve without self-intersections
There exist smooth - but not analytic - closed curves without self-intersections. I just would like to see a simple example of such a curve.
- 9
6
votes
4
answers
1k
views
What are the lengths that can be constructed with straightedge but without compass?
Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. ...
user2529
4
votes
1
answer
494
views
Is there a generalized Feuerbach point for an irregular non-Euclidean triangle?
Is the circle externally tangent to the three excircles of an irregular non-Euclidean triangle internally tangent to the incircle of the triangle, the tangent point being a generalized Feuerbach point?...
2
votes
1
answer
1k
views
Lebesgue covering dimension
Roughly from wikipedia: The covering dimension of a topological space $X$ is defined to be the minimum value of $n$ such that every finite open cover of $X$ has a finite open refinement in which no ...
- 6,034
5
votes
0
answers
581
views
When is polytope compatible with network flow?
A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
- 3,475
10
votes
2
answers
1k
views
Dense sphere packings which are not lattice packings
This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
- 4,015
6
votes
5
answers
4k
views
Formulas for equidistant curves
I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first ...
- 351
10
votes
1
answer
2k
views
Equations for an algebraic gömböc
A gömböc is a $3$-dimensional convex body (having uniform density) which has exactly one stable and one instable equilbrium position (see https://en.wikipedia.org/wiki/G%C3%B6mb%C3%B6c).
Such a convex ...
- 17.9k
5
votes
1
answer
817
views
Generalization of Moise's theorem
I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it.
The claim is ...
- 73
4
votes
1
answer
626
views
Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.
In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
- 59
4
votes
2
answers
682
views
Capacity of Balls in Hyperbolic Space
Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ...
- 3,626
16
votes
0
answers
763
views
Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces
Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.
Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
- 31.5k
6
votes
2
answers
2k
views
Are Pappus Theorems generalized?
Pappus' Centroid Theorems provide a slick way of computing the center of mass for plane curves and plane areas.
The first theorem states that the surface area $A$ of a surface of revolution generated ...
- 829
5
votes
2
answers
629
views
Approximate search space on a 5x5x5 cube with 3 different possible classes?
Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...
- 153