Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,405 questions
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Convergence in the Caratheodory sense and Hausdorff sense
Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence.
But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
- 239
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Two definitions of horofunction for Gromov hyperbolic spaces
Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, ...
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Does nonexpanding map between manifolds decrease volume?
(This question is a special case of a question I asked at SE, which got no answer there)
Let $M,N$ be diffeomorphic connected compact Riemannian manifolds, and let $f:M \to N$ be a surjective ...
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Lattice points on the boundary of an ellipse
How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...
- 1,072
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Fixed point of fatness
For the purposes of this question, define the following properties of convex sets in the plane:
A set is $R$-fat (for $R\geq 1$) if it contains a disc of side-length $x$ and is contained in a disc of ...
- 3,559
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314
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Longest induced cycles in random geometric graphs near criticality
We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...
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Restricted isometry
Here is a problem that has been bugging me for a while.
Let $\| \|$ be a norm over $\mathbb{R}^n$, let $C$ be a convex subset of $\mathbb{R}^n$ with non-empty interior, and let $f: (C,\|\|) \...
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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 ...
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Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
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Minimal surface enclosing two congruent balls
Let $B_1$ and $B_2$ be two unit-radius balls in $\mathbb{R}^3$ whose centers are separated
by a distance $d \ge 2$.
Q. For sufficiently small $d$,
is the minimal area surface enclosing $B_1$ and $B_2$...
- 151k
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Characterizations of metric trees
Let $X$ be a geodesic space. Then the following conditions are equivalent:
For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$.
No ...
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Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$
Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space?
A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder ...
- 42k
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Three homothetic centers are collinear
I am looking a proof for the problem as follows:
Let a convex hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their ...
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Questions on Thurston's metric on Teichmüller space
I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
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Isometry group of low dimensional Alexandrov space
It is known by the work of Galaz-García and Guijarro, that the dimension of the isometry group of an $n$-dimensional Alexandrov space (of curvature bounded below) is bounded above by $\frac{n(n+1)}{2}$...
- 561
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Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface
It is well known that
Grayson's dumbbell neck-pinch1,2 separates
into disconnected pieces under
mean curvature flow:
Image ...
- 151k
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Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
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About the surface area vs. volume of polytopes
Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter (...
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Point cloud that maximizes the minimum pairwise distance in Euclidean space
I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize
$min_{i \neq j} |...
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Rolling a sphere on a fractal curve
Given a rectifiable curve C : [0, 1] → R2 in the plane, it makes sense to roll a unit sphere S2 on the plane along that curve and ask what is its net rotation in SO(3).
I wonder if this also makes ...
- 2,858
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Geodesic line with endpoints in interior of Riemannian manifold or Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.
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- 21.8k
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Sizes of connected components from a random choice in a grid
This is inspired by the illustration in this recently updated question. So we take a (fairly big) $n$ and an $n \times n$ grid where we draw at random one diagonal in each of the $1 \times 1$ squares. ...
- 13.4k
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Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...
- 21.8k
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Smoothing of piecewise Euclidean Riemannian metrics
Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
- 14.4k
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From convex geometry to contact topology
Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine.
Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
- 13.5k
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Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
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Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
- 42k
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Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?
A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...
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The busy Star Guardian
On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
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Do compact universal covers have concentration of measure phenomenon?
$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their ...
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Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?
In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation:
For a metric space X they write $\mathcal{P}_1(X)$ ...
- 1,058
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Approximating a general rectangle partition by a guillotine partition
There is a rectangle $R$ partitioned into some axes-parallel rectangles:
The goal is to construct another partition of $R$ into rectangles, using only guillotine cuts.
That is: cut $R$ into two ...
- 3,559
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Variations on Gauss' trick
Cross-posted from MSE. This question is inspired by these two:
Non-trivial values of error function erf(x)?
Where is the mass of a hypercube?
Upon reading these two, I realized there might be a ...
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Metrically Ramsey ultrafilters
On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
- 42k
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Do manifolds with non-negative Ricci curvature allow bi-Lipschitz embeddings into Euclidean spaces?
QUESTION: Let $n$ be a natural number. Is it true that there exist $N(n), D(n) > 0$ such that any complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature can be embedded into $N$...
- 1,058
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Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
- 13.4k
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Hölder isoperimetric problem
Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the ...
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A conjecture on simplex
Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
- 705
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Geodesics between boundary points of a hyperbolic space
Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...
- 2,648
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Sharp isoperimetry in the discrete Heisenberg group
The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...
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Can the GUE be thought of as a uniform point in a high-dimensional polytope
I have thought about this question for a long time and could only find partial answers.
The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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Ricocheting pinball-like shot: Complexity?
Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...
- 151k
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Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
- 6,206
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How many facets can $\{\|D^T x\|_1\leq 1\}$ have?
$\newcommand{\RR}{\mathbb{R}}$Consider $x\in\RR^n$ and $D\in \RR^{n\times p}$ with $p\geq n$ and full rank. My question is:
How many facets can the polytope $ \{x\in\RR^n\ :\ \|D^T x\|_1\leq 1\}$ ...
- 12.7k
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Geometry of the metric cone
Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
- 13.5k
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Can two random graphs be metrically embedded into one another?
Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal ...
- 3,323
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Generalized flag complex?
Assume we glue an $n$-dimensional simplicial complex $K$
from copies of an $n$-simplex $\Delta$ with fixed spherical metric.
We may think that $\Delta$ has colored vertices
and we glue so that the ...
- 45k
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Systoles of hyperbolic (Riemann) surfaces of large genus
Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$.
The systolic inequality claims that for any ...
- 1,303
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438
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Gromov-Hausdorff and Lipschitz convergence of a non-collapsing sequence of manifolds with Ricci curvature bounded below
There is a theorem from Cheeger-Colding saying the following:
Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff ...
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315
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Minkowski's convex body theorem for ellipsoids
Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector.
Can this bound be improved ...
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