Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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A compact Alexandrov space with curvature bounded below has curvature bouneded above? [closed]
For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
- 471
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721
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understanding geometry of eigen values of Ricci tensor [closed]
As per I can visualize the eigen value $\lambda$ of a linear map $T:V \rightarrow V$, defined by $Tv=\lambda v$, is actually the scaling factor of the vector in the same direction as of $v$.My ...
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165
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Understanding the definition of an F-connected simplicial complex
I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...
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40
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Perimeters of the cells of a convex tessellation
Let $C$ be a compact, convex region in $\mathbb{R}^2$, and say we have scalars $a_i, b_i, c_i$ for $i\in\{1,\dots,n\}$. Consider the tessellation $R_1,\dots,R_n$ of $C$ defined by letting $$R_{i}=\{(...
- 4,049
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Inscribed polytopal approximation to a convex body
This question is on the continuation of the post
Approximation of convex body by polytopes
The central problem I am interested is an explicit construction of inscribed polytope with at most $n$ ...
- 599
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68
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How does the singular surfaces obtained when the border of a Euclidean set becomes a point look like?
I'm curious to understand in several manner, what is the metric geometry of the metric space homeomorphic to a sphere, obtained from a compact convex set $K\subset R^2$ with the Euclidean distance, ...
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Projection of a ray onto a random polytope
Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by $\operatorname{...
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Biggest volume parallelotope inside the union of two parallelotopes
Given a parallelotope $P$ symmetric around the origin, and a vector $v$, such that $(P+v)∩(P−v)$ is not empty, is there a simple way to obtain a parallelotope $Q⊂(P+v)∪(P−v)$, symmetric around the ...
- 244
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80
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A version of isotone projection cones
We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...
- 773
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262
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Virtually abelian centralizers
This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)
Anyway, I'm wondering what sort of groups have the ...
- 237
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70
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Covering number of the range of a function
I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
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114
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Shortest rope to capture a sphere of diameter 1 [duplicate]
I have a perfect rigid sphere of diameter 1.
I have infinite supply of rope. The rope is infinitely flexible and can be cut or glued without losing or adding length. The rope can be glued at any ...
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54
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Length invariance under nondecreasing changes of parameters
Suppose that $f\colon [0,1]\to [0,1]$ is a continuous, surjective and nondecreasing function, for example the Cantor function. Let $X$ be a metric space (not necessarily a length space) and let $L$ be ...
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86
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Projection from a polytope to an affine space
Let $P\subseteq \mathbf{R}^n$ be some polytope defined by an intersection of half spaces with corresponding hyperplanes $H_k$, and let $A\subseteq \mathbf{R}^n$
be some affine space, with $A\cap P \...
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109
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Intersection points of closed curves inscribed in a convex polygon
Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
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Quadrilaterals from a Unit Stick
This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):
Suppose one are given an ...
- 13.2k
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280
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A conjecture about cross sections of a pyramid [closed]
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
This is a conjecture I came across while trying to solve this problem.
I was ...
- 1,090
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93
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Coarse geometry of minimal surfaces in non-positively curved manifolds
Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface.
(You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
- 1,804
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74
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Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometry
Let $M$ be an open Riemannian surface of bounded geometry. Let $\Gamma$ be a group of diffeomorphisms of $M$. Suppose that $\Gamma$ is equi-quasi-isometric; i.e., its elements are (differentiable) ...
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Generators for the affine automorphism group of the octagon
Consider an octagon $O$ with opposite edges identified. Lemma 3.2.4 of (1) (subscription link) claims that the affine automorphism group of $O$ is generated by $D_8$ and the shear $\sigma$ such that, ...
- 881
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80
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Euclidean embedding of a graph based on 1-ring neighborhood distances only
Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...
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75
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Twisted calibrations and sufficient conditions on homology of sub-manifolds
I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
- 111
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156
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Entangled helical knots
Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...
- 151k
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60
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Finding special vectors generated by a matrix
Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
- 13.9k
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Covering a set of points by bounded geometric object/objects
1) Let $S$ be a set of $n$ points in $R^d$. Now, given a bounded geometric object $G$, the problem is to check whether $S$ can be contained in $G$.
2) Also, in general setting, the problem is to ...
- 285
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428
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Sampling a two-dimensional Gaussian distribution at points along an integer lattice
Please consider a two-dimensional Gaussian of the general form: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, where $C$ is the peak of the Gaussian, i.e. the point at which the ...
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Comparing two metrics on the space of infinite sequences and relating open and closed sets
Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...
- 798
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208
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Divergence of geodesics in $P(n,\mathbb{R})$
I'm reading Bridson & Haefliger's book on non-positively curved spaces. Specifically the parts in II.10 on $P(n,\mathbb{R})$. These are positive definite matrices, viewed as the image of symmetric ...
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Determining the position of a coordinate by binning Gaussian noise around that coordinate to lattice points with vertex-specific probabilities [closed]
(NOTE: I have changed and hopefully simplified this question by removing the section on randomly perturbing lattice points, and instead specifying that the counts at each vertex should be randomly ...
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203
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Does John's Ellipsoid preserve subset ordering? [duplicate]
Let $K \subset \mathbb{R}^d$ be a convex body, symmetric about the origin and with nonempty interior. Then John's theorem asserts that there exists a unique ellipsoid $E$ of minimal volume such that $...
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262
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Is the size of $\varepsilon$-nets of the Euclidean ball exponential for large $\varepsilon$
Let $X$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_2)$. A finite set $N=N(\varepsilon)$ is a $\varepsilon$-net of $X$ if every point in $X$ is at most a distance $\varepsilon$ from a point from $N$.
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A few questions about Banach spaces all of whose "points" are infinite sequences of real numbers
I shall call these spaces "sequence spaces". While there exist well known separable Banach spaces that have subsets which are isometric to every separable metric space, I never heard of any separable ...
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Equidissection of square [duplicate]
Monsky's Theorem states:
One cannot dissect a square into an uneven number of triangles with equal area.
I was wondering how close one could get to a equidissection, i.e.
For $n$ an uneven number,...
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Alexandrov spaces satisty $BE(K,N)$ and $BE(K,\infty)$?
Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi -(\Gamma(f,Lg)+\Gamma(...
- 689
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curvature of subset of Alexandrov spaces
If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$ds_M^2 = d{t^2} + \exp \left( {2t} \right)ds_N^2$$ N is a submanifold of M. Then by Gauss-equation, we can prove $Ric\left( N \...
- 689
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565
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The proof of Cheeger's splitting theorem for almost nonnegative Ricci curvature manifolds
Cheeger's splitting theorem says "Let $\left( {M_i^n,{p_i}} \right)\mathop \to \limits^{G - H} \left( {X,p} \right)$ with $Ric\left( {{M_i}} \right) \ge - \left( {n - 1} \right){\varepsilon _i}$ $\...
- 689
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Covering the annulus of symmetric convex body
Consider a symmetric convex body $A$ in $\mathbb{R}^d$. Now, we draw another object, $A'$, concentric and translated with respect to $A$ and having radius slightly greater than twice to the radius of ...
- 285
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179
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Boundary surfaces in a 3d Voronoi tessellation with obstacles
Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...
- 1,125
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Uniformly sampling the solution space for points where the free termini of two rays, anchored at 3-space points, can intersect
I have two rays, one of length $L_1$ and one of length $L_2$. I anchor these rays, each at one end, on the 3-space points $p_1$ and $p_2$. Assuming that the Euclidean distance between $p_1$ and $p_2$...
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142
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Relationship between stabilizers of a general point and a boundary point
Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...
- 105
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300
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Pencils of circles and Liouville's theorem
Is there any relation (maybe implicit) between the conformal geometry in the space of circles and spheres and the study of harmonic functions?
In the original question I was musing whether the ...
- 13.5k
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392
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Frustrating the number of possible common edges between two connected components composed of square Penrose tiles
Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...
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Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
- 141
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185
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Compute generalized pentagram map
Hi,
(This is my first question on MathOverflow! :-)
Imagine you have a set of points $S = \{p_1, \ldots, p_n\}$ in $\mathbb{R}^d$, of which $t$ are "bad". I want to compute a "safe convex hull", ...
- 11
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167
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How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?
Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$.
Given a general 4-...
- 119
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247
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dissections and vertices of non-convex polytopes
Let us call a finite union $P$ of $n$-dimensional compact convex polytopes in $\mathbb{R}^n$ a non-convex polytope. Recall that a dissection of $P$ is a finite collection $T$ of $n$-dimensional ...
- 14k
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371
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Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
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526
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Discrete subgroups of isometry group of proper metric space
Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.
Consider the following ...
- 13
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385
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General setting for triangle inequalities (terminology question)
Regarding the "mathematical object $(X, s)$" described below (in general, or under some more specific conditions) I'd like to know whether it is called a particular name, for reference in the ...
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128
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Proving that an optimal solution "converges"
This question is a follow-up on a previous question I asked at:
Distances between and among points in a region
Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...
- 195