Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,263
questions
6
votes
1
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498
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Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity
Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...
- 7,087
13
votes
2
answers
1k
views
Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
- 149k
9
votes
2
answers
899
views
Shortest irrational path
What is the shortest curve $\gamma$ in $\mathbb{R}^2$
from the origin $o=(0,0)$ to a rational point $p=(a,b)$
that (a) passes through no other rational point, and
(b) contains no point a ...
- 149k
2
votes
0
answers
275
views
Show a Map Defined on $S_3$ (trivially-embedded) in S^4 extends.
Hi, Again:
I am trying to understand an argument I wrote down a long time ago, to show that a given
element of $M_3$ (mapping-class group of a) genus-3-surface, defined on a trivially-embedded copy ...
- 105
7
votes
2
answers
335
views
non-rigidity of interior points in polyhedral triangulations?
It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (...
- 13.7k
1
vote
2
answers
384
views
Ahlfors' proof of locally K-quasiconformal to K-quasiconformal
This is a question I originally posted in Math Stack Exchange, but perhaps the question was too specialized, so I thought I'd post it here instead
I'm currently reading through "Lectures on ...
- 235
7
votes
2
answers
763
views
Shortest paths on linked tori
I will make this question specific at first, and general later.
Suppose we have two linked tori, $T_1$ and $T_2$,
each of radii $(2,1)$, meaning that each torus is the result of sweeping
a circle of ...
- 149k
5
votes
2
answers
630
views
Shortest "painting" of the sphere
Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing ...
- 1,513
7
votes
0
answers
315
views
Erlangen program carried out explicitely?
I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations.
What I mean by "carrying out Erlangen program" is picking a specific group (say SO(...
- 71
3
votes
1
answer
554
views
A question about Jung's theorem
A theorem of Jung states that, given n pairwise distinct points in the Euclidean plane E, there is a
unique circle of smallest radius in E that contains all the points and its radius is less than or ...
- 6,105
1
vote
1
answer
574
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Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges
Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.
Motivation -
I'm interested in a particular case of the problem where one wants to ...
1
vote
1
answer
281
views
Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges
I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:
(1) - I am only allowed to use a construction block, $v_i$, consisting of a single ...
100
votes
6
answers
5k
views
Light rays bouncing in twisted tubes
Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...
- 149k
8
votes
1
answer
785
views
The rain hull and the rain ridge
Rain falls steadily on an island, a 2-manifold $M$, which you may
assume, as you prefer,
is: (a) smooth, or (b) a PL-manifold, or perhaps even
(c) a
triangulated irregular network (TIN).
After a time,...
- 149k
15
votes
1
answer
2k
views
Ping-pong relief map of a given function z=f(x,y)
I have an idea to design a type of
Galton's Board
to "draw" a relief map of a given two-dimensional function $z=f(x,y)$.
A typical Galton's Board drops, say, ping-pong balls through a series
of evenly ...
- 149k
0
votes
0
answers
563
views
Pixel Approximation of Circle
Dunno whether this is a good problem for MO anyway. Well, here it goes:
On a screen you have only little squares, so a natural question is
which n-omino approximates a circle best. The problem is less ...
- 4,699
1
vote
1
answer
140
views
Inferring geometric properties of a polytope from intersection volumes of spheres at unknown coordinates on its surface
Let's say we have some polytope $P$ in 3-space (which is not necessarily convex) as well as some number of points on its surface, $(g_1, ..., g_N)$. We are provided no information about the ...
- 309
8
votes
1
answer
623
views
Name of a metric space concept
I have a metric space with the following property (a bit like having unique geodesics): for any points $a,b,x,y$ with $d(a,b)=d(a,x)+d(x,b)=d(a,y)+d(y,b)$ and $d(a,x)=d(a,y)$, we have $x=y$. Is there ...
- 55k
64
votes
6
answers
5k
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Shortest closed curve to inspect a sphere
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and
exterior to $S$
which has the property that every point $x$ on $S$ is visible to some point $y$ of $...
- 149k
3
votes
0
answers
470
views
Higher order Pansu derivative
Given a group $(G,*)$ there is no candidate for what can be understood as a derivative of a function $$f:G\rightarrow\mathbb{R}.$$ However, for the special case of Carnot groups there is the so-...
- 31
8
votes
2
answers
1k
views
Maximal number of connected components of complement to an affine plane real algebraic curve
Let $X$ be a (singular, reducible) affine plane real algebraic curve of degree $d$.
How we can estimate maximal number of connected components of it's complement in $R^2$ in terms of degree?
- 403
25
votes
2
answers
1k
views
Geometry of complex elliptic curves
Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat?
- 1,387
0
votes
2
answers
376
views
Mean Three Dimensional Shape of Surfaces
If I have $n, 1 < i < n, $ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface?
(I'm unsure what I mean by average - intuitively it's obvious, ...
- 1
4
votes
4
answers
710
views
Parametrizing the realization space of a polyhedron by its edges
I alluded to this here, but at that point I hadn't really done enough work to know what I wanted to ask.
Call a polyhedron "trihedral" if three faces meet at each vertex. Each of the F faces can be ...
- 3,522
4
votes
1
answer
446
views
When can a 3-dimensional triangulation be isometricaly embedded in R^n?
Consider a triangulation of some bounded region of $R^3$ with a (finite) set of tetrahedra (like in Regge calculus). It can be thought of as a simplicial 3-complex with specified lengths of edges. The ...
- 221
4
votes
1
answer
256
views
Polar interpretation of convexity
Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
- 635
6
votes
2
answers
633
views
A convex polyhedral analog of the pentagram map
I am wondering if there is a three-dimensional analog of
the pentagram map, which maps a convex polygon to another
convex polygon. Here's the Wikipedia image:
I am seeking something similar that maps ...
- 149k
1
vote
1
answer
170
views
Is there a linear embedding of a simplical 3-complex in R^6?
I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...
- 221
3
votes
1
answer
300
views
Average squared distance in $k$-regular graphs
Let $X=(V,E)$ be a finite, connected, $k$-regular graph. Let $avg(d^2)$ be the averaged square distance between vertices, as defined in Average squared distance vs diameter in vertex-transitive graphs ...
- 11.1k
4
votes
1
answer
3k
views
intersection of convex and non-convex polyhedra
I am trying to find the best appropriate way to intersect polyhedra which may be non-convex.
The number of vertices that build the polyhedron is hence always small (up to 20 or so).
The ...
- 143
4
votes
1
answer
339
views
A question about bisecting plane convex sets
Suppose that S is a compact convex subset of the Euclidean plane E whose interior is non-empty.
If p is a point of E such that every straight line in E which passes through p bisects the area
of S, is ...
- 6,105
6
votes
3
answers
1k
views
Average squared distance vs diameter in vertex-transitive graphs
Let $X=(V,E)$ be a finite, connected graph on $n$ vertices, endowed with its graph metric $d$. The average squared distance of $X$ is $avg(d^2)=\frac{1}{n(n-1)}\sum_{x,y\in V,x\neq y} d(x,y)^2$; it ...
- 11.1k
1
vote
0
answers
2k
views
Fitting an ellipse to an arbitrary polygon
Hello,
I'd like an algorithm for fitting an ellipse to a polygon. This polygon may be convex or concave. I've read about fitting an ellipse inside or outside a polygon (maximal and minimal of size, ...
- 11
16
votes
4
answers
3k
views
covering by spherical caps
Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius $\...
- 95.6k
9
votes
2
answers
852
views
Concentration of measure for arbitrary convex bodies?
There are various "concentration-of-measure" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an $\...
- 149k
3
votes
2
answers
692
views
Sets invariant under sections
Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$:
$$
s(x) = (y\in X|\langle x,y \rangle \in \bar{G})....
- 1,605
10
votes
1
answer
905
views
Metrically singular Alexandrov space.
Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf ...
- 4,033
8
votes
4
answers
1k
views
Shortest Path in Plane
I thought about the following problem:
Given a polygonal subdivision of the euclidian plane where each of the polygons has a speed associated with it, and given two points s,t, I'm interested in the ...
- 239
3
votes
2
answers
710
views
Simultaneous resolutions and deformations of simple singularities
Let $X\to \Delta$ be a flat family of complex surfaces with at most a finite number of singularities of simple type, where $\Delta$ is a complex domain in $\mathbb C$.
Here simple type means ...
- 289
14
votes
0
answers
548
views
Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 12.3k
6
votes
1
answer
424
views
A toy model for the t-section problem
Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H $ is some prescribed function (small, smooth, vanishing near the endpoints to any ...
- 59.8k
2
votes
1
answer
382
views
Feasible space of SDP
Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible ...
- 505
3
votes
2
answers
3k
views
Parametrization of the intersection of an ellipsoid with a sphere
First I would like to say that geometry is far away from my domain.
I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy problem.
So ...
- 1,304
2
votes
1
answer
897
views
A density condition for metric spaces
I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?
Property: $(X,d)$ metric space. For ...
- 5,904
4
votes
1
answer
3k
views
complex gradient of a function
Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...
- 41
12
votes
3
answers
992
views
F→E→B bundle with B,E,F hyperbolic: possible?
It would be interesting to me obtain an answer to the following easy to state question:
Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...
- 579
1
vote
2
answers
328
views
Can a set of tetrahedra glued together by a common vertex be isometrically embedded in R^4?
A collection of triangles with a common vertex $A_1VA_2$, $A_2VA_3$, ... $A_NVA_1$ with specified side lengths can be isometrically embedded in $R^2$ provided the angles around $V$ add up to $2\pi$. ...
- 221