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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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 ...
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What is the definition of product of ideal sheaves?
Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I
is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right?
Thanks.
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Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
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Question on constraints
Does there exist any set of 6 real numbers $a_{ij}$ for $1 \leq i < j \leq 4$ satisfying the following conditions:
\begin{aligned}
&0 \le a_{ij} \le \pi, \\
&a_{ik} + a_{jk} > a_{ij}, \\
...
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Can either pair of opposite sides of an arbitrary parallelogram be brought into coincidence isometrically in 3-space?
Let P denote any nondegenerate planar parallogram, and let A and B be either pair of its opposite edges.
Does there always exist a continuous family of locally-isometric mappings ht of P into 3-space, ...
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Connectedness of fibers of almost Riemannian submersions
EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary.
Given $f\colon M\to N$ be an $\varepsilon$-...
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To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region
We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above ...
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Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
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To optimally wrap convex laminae with paper
Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra
Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ...
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Defining area / n-volume of a finite metric space
Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested ...
- 820
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What is the locus of points from which a given segment AB subtends an angle of alpha in non euclidean geometries?
In euclidean geometry, the locus of points from which a given segment AB subtends a given angle of alpha is made up of two arcs (https://pt.wikipedia.org/wiki/Par_de_arcos_capazes). My question is ...
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127
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Smallest trapeziums containing a given convex n-gon
Question: Given a planar convex $n$-gon $C$, to find the smallest area / smallest perimeter trapezium (trapezoid) - a convex quadrilateral with at least one pair of mutually parallel edges - that ...
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$1$-Lipschitz map from hyperbolic to Euclidean plane
I'm trying to find a reference to the following statement.
Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar ...
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Isolated maxima for sum of distances of points on a manifold
Let $X$ be a closed Riemannian manifold and consider the function $f_n : X \times \cdots \times X \to \mathbb{R}$ where the domain of $f_n$ is the $n$-fold cartesian product of $X$ and where $f_n(p_1,....
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How to reduce the compact support to the case of small diameters in Tao's "A sharp bilinear restriction estimate for paraboloids"
I am reading Terence Tao's paper "A sharp bilinear restriction estimate for paraboloids"
to prove the bilinear restriction estimate on paraboloids. In Section 3, he assumes that $\text{diam}(...
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A 'natural' enumerable metric space with integral distances which is essentially the Euclidean space
It is easy to construct a metric space $E_d$ such that all points
of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space ...
- 17.9k
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What is the limit of a helix as the frequency tends to infinity?
Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$
My initial ...
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Are simplicial polytopes a dense set?
Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric.
Are simplicial polytopes a dense subset of that space?
Probably this is just a ...
user467453
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Packing number in finite-dimensional normed spaces
I am working on a paper and quoted the following result from these lecture notes.
Where can I find a reference to this result either in a book or a paper, that I can cite?
(I looked on the course ...
- 5,405
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Dimension-preserving non-linear map
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\...
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A neighborhood $Y$ of a set $X$ such that the line segment connecting any point in $Y$ and its projection to $X$ is contained in $Y$
A direct line from a point $p$ to a set $X$ is a line segment with one endpoint at $p$ and one endpoint in $X$, which is as short as any other line segment from $p$ to $X$. Given a closed set $X$ and ...
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When are uniform embeddings quasisymetric
Let $X,Y$ be metric space and suppose that $f:X\rightarrow Y$ is a uniform embedding; i.e.:
$$
\omega(d_X(x,z))\leq d_Y(f(x),f(z)) \leq \Omega(d_X(x,z)),
$$
where $\omega\leq \Omega$ are both strictly ...
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Characterization of greedy TSPs?
Define a greedy tour of a set $S=\{p_1,\ldots,p_n\}$ of $n$ points in $\mathbb{R}^2$
as produced by selecting the $i$-th point $p_i$ to start, and then connecting to the nearest neighbor $p_j$ to $p_i$...
- 151k
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Conditions for Lipschitzness of boundary normal vector, almost everywhere
Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
- 6,853
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428
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Covering number in the space of symmetric matrices
Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|\cdot\|_F$ the Froebenius norm. Consider ...
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Are there any applications of linear algebra over the complex numbers, where the role of complex conjugation is replaced with the trivial involution?
The complex inner product $\langle u, v \rangle$ is a special case of a sesquilinear form over a field. Its definition is $\langle u, v \rangle = \sum_{i} u_i \overline{v_i}$. There is clearly the ...
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Asymptotic cone
Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
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Geodesic triangles and Gauss curvature of induced metric
Let $V$ be a complete Riemannian manifold with sectional curvature $K(V) \leq -1$. Let $\Delta^2 \subset \overline{V}$ be a geodesic triangle in its universal cover. When is it true that the Gaussian ...
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Apartment in non-discrete Euclidean building with prescribed properties
Let $X$ be a non-discrete Euclidean building. Let $x \in X$, $\Delta_x$ be the germ of a Weyl-chamber based at $x$ and $\xi$ be a point at infinity. Choose $y \in \Delta_x$.
Is there an apartment ...
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On triangulations and "coverage" of circumcircles
Let $P$ be a convex quadrilateral defined by four vertices $a$, $b$, $c$, and $d$. Suppose that the circumcircle of $\triangle abd$ contains $c$.* Let $D(\triangle abc)$ to denote the area enclosed by ...
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Thirteen-point conic and four-point line, are they new?
We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
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Stability of isoperimetric inequality
Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1.
Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$,
where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...
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Least square assignment and hyperplanes
Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ ...
user178738
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Modulus of continuity of parameterizing Wasserstein
Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported ...
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A formula for the area of bicentric quadrilateral
Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...
- 2,661
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Vertices of 2 self-polar triangles lie on conic
I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?
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Is this relation between planar convex hulls and heaviest cliques true?
If $P$ is a set of $n$ points in the euclidean plane whose convex hull $\operatorname{CH}(P)$ has $h$ corners, and $Q\subset P$ has $m\le\lfloor\frac{h}{2}\rfloor$ points and maximal sum of pairwise ...
- 13.2k
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How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?
In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram:
\...
- 6,201
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Exact volume calculation of a polytope is NP hard under which restrictions?
Computing the exact volume of a polytope given in half space representation seems to be NP-hard. One paper I found proved it is hard for rational coefficients. (However, the paper itself was behind a ...
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Terminology: Co-completion of Met?
In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...
- 5,405
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Lattice points in hypercubes
Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
- 503
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743
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tetrahedral interpolation and integration along a segment
Let's say we have a several tetrahedrons $T_i$ whose faces touch so that each face belong to two tetrahedrons. Each tetrahedron contain a value $V_{i}$.
Given a position $P$ inside the tetrahedron $...
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Distance pairs in labeled directed graph
Suppose we have a simple directed graph with $n$ nodes and $m$ edges, and we label each edge from $1$ to $m$ (with distinct labels). Define the weighted "length" of a directed path to be the maximum ...
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Triangles with a given outer Soddy circle of the Malfatti circles
I did a JavaScript interactive picture of the Malfatti circles of a triangle. The user can drag the vertices of the triangle and the Malfatti circles are updated accordingly.
Now, I would like to ...
- 2,560
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Lipschitz vs. bi-Lipschitz parametrizations for subsets of Euclidean space [closed]
Let $n \in \mathbb{N}$. Is there a standard example of a subset of $\mathbb{R}^{n+1}$ that is contained in the image of a Lipschitz map $\mathbb{R}^n \to \mathbb{R}^{n+1}$ (or, more generally, that is ...
- 565
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Embedding a graph in $\mathbb{R}^3$ with partial geometric information
I have a connected, sparse, graph (a molecule to be specific) and I'm interested in associating 3D coordinates with the vertices. Here's the kicker: I already have coordinates for none/some/all ...
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313
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Group action on quasi-isometric geodesic metric space [closed]
If a group $G$ acts on a geodesic metric space $X$, then does $G$ act on a geodesic metric space $Y$ which is quasi-isometric to $X$?
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Is volume of abstract polytope realisation bounded by length of edges?
Suppose we have abstract polytope $F$ of dimension $d$ (that is the greatest rank facet has rank $n$). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$,...
- 1,626
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Are there polygonal tilings with infinitely many positions, each (or at least one) occurring infinitely often?
My recent question about polygonal tilings where tiles can occur in infinitely many positions has been answered by two nice constructions (besides Jan Kyncl's answer, there is the Conway tessellation ...
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Small codimension 1 ball on the boundary of metric ball in Busemann G-spaces
Let $(X,d)$ be a metric space. $X$ is said to be a Busemann $G$-space provided it satisfies the following axioms:
(1) Menger Convexity: Given distinct points $x,y\in X$, there is a point $z\in X-\{x,...
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