Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,404 questions
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Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width? [closed]
Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have?
Thank you!
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Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective?
Suppose X is a proper Gromov hyperbolic space and $\partial X$ is its Gromov boundary. It is well-known that there is a canonical group homomorphism $\Phi$ from the isometry group of X to the group ...
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Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
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Symmetry of infinite locally finite trees
Let $\Gamma$ be an infinite, locally finite tree without end points, with the additional property that there exists a positive integer $N$ such that for each occurring valency $v$, we have that $v \...
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Minimal surface enclosing balls
(This question is tangentially related to an earlier
question I posed: Minimal surface enclosing two congruent balls.)
Let $B_1,\ldots,B_k$ be unit-radius balls in $\mathbb{R}^3$, with pairwise
...
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When are Wasserstein spaces $CAT(\kappa)$?
Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...
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Effect of snowflaking on doubling constants
This question is related to this one. Let $(X,d)$ be a metric space, let $\epsilon\in [0,1)$ and consider the snowflake $(X,d^{1-\epsilon})$. Suppose that $(X,d)$ has a finite doubling constant, ...
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On a possible variant of Monsky's theorem
See Wikipedia for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.
Questions: Are there quadrilaterals that allow partition into ...
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Could somebody suggest a way to determine if a parallelogram contains another parallelogram?
I thought of one way to do this.
Using the algorithm which determines if a point is inside a parallelogram,
one can determine if the polygon contains the point within $2N$ steps ($N=2$ for ...
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A generalization of Harcourt's theorem
This question is closely related to my previous question.
Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem.
Claim. Let $A_1,A_2 \ldots ...
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Geometric sampling problem in the Euclidean space in high dimensions
Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...
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Upper bounds for high-dimensional spherical codes given the covering radius
I assume that this sort of question has already been considered at great length. Nevertheless, I could not find an answer to this question in the related literature.
Given a constant $a\in (0,2]$, ...
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Mapping of subcubes of a $(d+k)$-hypercube onto subcubes of a $d$-hypercube
Denote by $Q_n$ the $n$-dimensional hypercube. A vertex of $Q_n$ is represented by a vector of $n$ $\{0,1\}$-bits. An edge corresponding to two vertices whose vectors differ in one coordinate is ...
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How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain?
1) How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain (Volume)?
2) How many nodes $v$ does a ball with center $x$ of radius $r$ in the Johnson graph $J(n,k)$ contain such ...
user6671
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Repeatedly halve and twist a planar shape: Limiting shape?
Consider the following iterative process.
Start with a planar region $R=R_0$ of $\mathbb{R}^2$.
I am thinking of $R$ as connected,
but it may become disconnected.
In the example below, $R$ starts as ...
- 151k
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Characterizing 1-ended graphs
I just came across the notion of ends of a space, and I wonder if the following are equivalent for $G$ a locally finite connected graph:
There exists an infinite path $v_1,v_2,\dots$ in $G$ which ...
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Pairwise distance distribution for point clouds (normal distribution) [closed]
I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$).
My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
user76454
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680
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Regular paths along surface of sphere
I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$.
The movement should be repetitive, so that ...
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How to construct a harmonic function with non-zero gradient on manifold with two nonparabolic ends?
We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole $M$.
More ...
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Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
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General and translational Birkhoff lattices. Equational classes
By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
Is there an equational class ...
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Covering a $d$-dimensional integer lattice by repeating a minimal set of deterministic moves
Imagine I place a turtle on some desired vertex, $v_i$, of a bounded $d$-dimensional integer lattice, $Z^d$, with dimensions $(l_1, ..., l_d)$. The turtle is able to travel from vertex to vertex ...
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Distance calculation in metric space
Dear All,
I want to calculate the distance between two sets in which the maximum distance between the sets are minimized. Formally problem defined as,
$\displaystyle \min_{a \in A} \max_{b \in B}$ d(...
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Covering an arbitrary polygon with minimum number of squares
I have a problem whereby, given an arbitrary polygon with any number of points, I need to cover the whole area by a number of fixed size squares. I can easily find a set of squares which covers the ...
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Tangent lines to 2 circles, tangent planes to 3 spheres, and so on.
Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them:
Given two circles in the plane, there is (at least) a line which is tangent ...
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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 ...
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Synthetic Proof for Ratio of Volumes of Concentric Spheres?
Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following.
$(1)$ Show that $$ \lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}...
- 23k
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Settling a circular argument: room for one more?
By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller ...
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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 ...
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Systems of conics
It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
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Various Cartan's Lemmata
I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
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Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
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Proving non-existence of non-frictional CVTs?
This is a bit of a weird question because the problem is more about how you could even go about formalizing a hypothesis more than how to prove it — but it seemed like a fun idea and I figured someone ...
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Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
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Fitting a simplex to set of almost orthogonal vectors
I am trying to solve the following question, that I guess (hope?) has been solved before but I couldn't find any reference.
Let $S$ be a set of $d$ unit vectors in a $d$-dimensional Euclidean space ...
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A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
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Local Lipschitz constant of exponential map for Hadamard manifolds
Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$. Let $r>0$ and fix any point $p\in M$. Is there a bound on the local ...
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Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
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Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum [closed]
Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?
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Closed surfaces of prescribed mean curvature
Let $D\subset\mathbb R^n$ be a smoothly bounded open domain and $0\in D$. For any $x\in\partial D$ there holds
\begin{eqnarray*}
2 \,a'(\vert x\vert)\,(x\cdot\nu(x))+(n-1)\,a(\vert x\vert) \, H(x) = \...
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Equal products of triangle areas
Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
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Generalizing Bottema's theorem
Can you provide another proof for the claim given below?
Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
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On convex polygons contained in convex polygons
In what follows '$n$-gon' stands for '$n$-vertex polygonal region'.
Question: Given a convex $n$-gon $C$, find the smallest convex region $R$ such that $C$ is the smallest $n$-gon that contains it.
...
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Surjective affine mapping from the n-cube to a regular convex polygon with k^n vertices
I am wondering if there exists a surjective affine mapping from the n-cube to a regular convex polygon with k^n vertices (for any k? maybe just some k?). As an example I tried to think of a surjective ...
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Cross ratio in hyperbolic geometry
In the rough sketch four concurrent lines are drawn in the Poincaré disk model and in the Euclidean model.
If same angles $ (\alpha,\beta,\gamma,\delta) $ are enclosed at respective points of ...
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Inferring the modulus of continuity
Let $f:X\rightarrow Y$, $g:Y\rightarrow Z$ be uniformly continuous functions between metric spaces $X,Y,Z$ with moduli of continuity $\omega_f$ and $\omega_g$, respectively. Suppose that we know that ...
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Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
- 5,405
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Property of convex polygons on integer lattice structures
Another graduate student and I are working on an research project and are looking for a paper or other source that has a proof for a result about polygons on an integer lattice structure. Suppose you ...
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Wasserstein space with strictly non-positive sectional curvature
Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$.
Does it ...
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