Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

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find a parametric equation for the that is perpendicular to the graph of the given equation at the given point: z=x3-xy2 , (-1,1,0) [closed]

i am still a beginner in this and don't know how to do it so please can you help? find a parametric equation for the line that is perpendicular to the graph of the given equation at the given point: z=...
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78 views

2 dimensional Hausdorff measure and area measure on the hyperbolic plane

$\newcommand{\diam}{\operatorname{diam}}$From the Encyclopedia of Mathematics, the Hausdorff measure on a generic metric space (X,d) can be defined using $H^\alpha_\delta (E):=\omega_\alpha \inf \{\...
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39 views

Background smooth geometry vs background edge geometry

In this note, page 8 and page 9, the Kähler–Einstein edge equation is written first in (3.7) in terms of a model edge metric and in (3.11) in terms of a smooth Kähler metric. What is the distinction ...
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71 views

Distance between points on two geodesics starting from the same point in Gromov-hyperbolic space

I was going through the book "Metric Spaces of Non-Positive Curvature" by Bridson and Haefliger and got stuck while proving the following Lemma: Lemma 1.15: Let $X$ be a geodesic space that ...
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Is the Haversine function a norm? [closed]

We're working on some tooling at work and we're curious if it's accurate to include the Haversine function in our norm package with Euclidean and Manhattan/Taxicab (see also WP on the Haversine ...
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84 views

Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric. I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
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1answer
66 views

How to fold a tesseract from L-unfolding?

I came across an image, that show really simple unfold of 4-dim cube. https://arxiv.org/pdf/1512.02086.pdf here at #2.1, and 120 here https://page.mi.fu-berlin.de/moritz/mo/198722/unfoldings.html. ...
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107 views

How to correctly state Cauchy's rigidity theorem?

Cauchy's rigidity theorem is usually cites briefly as Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent. As a more formal generalization to general ...
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68 views

The area of cross-section of polyhedron or polytope

Consider a polyhedron or polytope $P$ in $\mathbb{R}^n$. Let $F$ be a bounded face of $P$. Now we shift and rotate this polyhedron such that $F$ lies on the hyperplane $\{t=0\}$, and $P\subseteq\...
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Normal from a point to paraboloid [closed]

Prove that the normals from the point ( α,β,γ ) to the paraboloid $x^2/a^2 + y^2/b^2 = 2z$ lie on the cone $α/(x-α) + β/(y-β) + (a^2 -b^2)/(z-γ) = 0$
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56 views

Which polytopes can be deformed while keeping their edge-lengths?

Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while keeping its combinatorial type, and keeping its ...
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1answer
40 views

Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?

Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
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1answer
63 views

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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Bound for the cardinality of maximal $r$-separable subsets contained in a ball of radius $R$ in $\mathbb R^d$

Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that ...
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1answer
230 views

A geometric approach to the odd perfect number problem?

Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$. Let $h(n) = J_2(n)$ be the second Jordan totient function. Define: $$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)}...
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143 views

Estimate of the side length of a small triangle

Let $M$ be a Riemann surface with Gauss curvature $K \le 1$. Let $\gamma_1$ and $\gamma_2$ be two geodesics starting from a fixed point $p \in M$. We denote the other endpoints of $\gamma_1$ and $\...
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46 views

Is the mean center of the vertices of a convex polygon always inside the polygon? [closed]

As simple as that. I'm doing an R program where I need to order clockwise a bunch of points that describe a regular polygon and to do that I figured I could find a point inside, change to polar ...
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1answer
73 views

Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
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70 views

Recursive expression of Lebesgue measure for balls with removed intersection

This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known). Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
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1answer
552 views

Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that all vertices end up on a common sphere, and the ...
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2answers
157 views

Contractible Rips complex from non-hyperbolic group

I heard that the Rips complexes associated to the Cayley graphs of hyperbolic groups are contractible for a sufficiently large radius. Is the converse true? Namely, if a group is non-hyperbolic, then ...
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What practically computable homotopy and/or (co)homology theories are known for finite (di)graphs, metric spaces, etc?

Of late I have taken to applying Dowker homology and the path homology theory of Grigor'yan et al. like a hammer to various relations and/or digraphs that have looked like nails. At the same time, I ...
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88 views

The space of $p$-adic norms

The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...
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Does the lemma remain valid in b-metric space?

Let $(X,d)$ be a complete metric space. $$CB(X)=\{A : A \text{ is a nonempty closed and bounded subset of }X \},$$ $$D(A,B)=\inf \{d(a,b) : a\in A , b\in B\},$$ $$\sigma (A,B)=\sup \{d(a,b) : a\in A , ...
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1answer
128 views

Geometric construction of the fourth intersection points of two conics

In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c_1$ be the conic passing through those three points and $D_1,E_1$. Let $...
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3answers
90 views

Can you measure the degree of uniformity of a 2d shape?

Is there a calculation that could take the points that make of the outline of a 2 dimensional shape and provide a numeric evaluation representative of the uniformity or symmetry of the shape. Such as ...
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42 views

Generalized quadrangles and their connection to prime powers

Generalized quadrangles are a commonly know geometric structures. A generalized quadrangle is an incidence structure $(P,B,I)$, with $I \subseteq P \times B$ an incidence relation, satisfying certain ...
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2answers
108 views

Fast algorithms for calculating the distance between measures on finite ultrametric spaces

Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula $$\...
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1answer
32 views

What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?

I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
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1answer
94 views

Packing in uniform domains

Given $N$ points $X:=(x_i)_{i \in \{1,..,N\}}$, we now define a score function $S:X \rightarrow \mathbb{N}$ that is $S(X)= \sum_{i=1}^N S(x_i)$ where the score of $S(x_i)$ is $$S(x_i) = 2* \vert \{x_j;...
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3answers
228 views

Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope. Suppose that I know its combinatorial type (aka. the face-lattice), the length $\ell_i$ of each edge, and the distance $r_i$ of each vertex from the ...
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1answer
221 views

The space of skew-symmetric orthogonal matrices

Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \...
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33 views

Lax CD(K, $\infty)$ space in the sense of Sturm

In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\...
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56 views

Area of minimising surface

I am interested in calculating the least area of a surface spanning the boundary of an octant on the unit sphere; and short of precise values I am looking for upper bounds for this area. In $\mathbf{S}...
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1answer
200 views

Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here: I am trying to understand the article by Maryam ...
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1answer
65 views

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, ...
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164 views

Diameter comparison for manifolds with $\text{Ric}\ge 0$

Let $M$ be a Riemannian manifold with $\text{Sec}\ge 0$. From Topogonov Theorem follows that for every $p \in M$ the quantity $$ \frac{\text{Diam}(B_r(p))}{r} $$ is non-increasing in $(0,\infty)$. ...
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90 views

Homogeneous metric surfaces

I am looking for a reference for this result. Let $S$ be a metric space such that it is homeomorphic to a two-dimensional manifold, it is 2-homogeneous: given two pairs of points $(x,y)$ and $(x',y')...
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89 views

Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes

Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement? There exists ...
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37 views

Neighbor count in sphere packing in N dimensions

So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" ...
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1answer
125 views

Can prolates overlap more easily than oblates?

Context: When modeling anisotropic particles, the two common types of shapes of interest are cylindrical and disk-like particles. For simplicity let us say we model these as prolates and oblates ...
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Klein geometry associated to a degenerate conic

In his study of non-euclidean geometries, Felix Klein considers the group of projective transformations acting on the real projective plane whose extensions to the complex projective plane preserve a ...
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1answer
97 views

Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
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97 views

Two questions on counterexamples to Borsuk's conjecture and ball-packings

In 1933 Karol Borsuk conjectured the following Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$? Whilst new to this ...
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1answer
87 views

Is Sydler's theorem concerning Dehn invariants constructive?

Sydler proved something of a converse to Dehn's negative resolution of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that "every two Euclidean polyhedra with the same volumes and Dehn ...
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136 views

Applications of the co-area formula

Kirchheim [2] generalized the classical area formula to the case of Lipschitz mappings into metric spaces. Ths paper is well known and widely cited. The area formula is a special case of the co-area ...
3
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1answer
247 views

number of integer points inside a triangle and its area

Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the ...
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37 views

If the volume-ratio of an inscribed convex set to the circumscribing convex set is rational, can anything of consequence be further deduced?

Say, one has two $n$-dimensional convex sets $A$ and $B$, with $B$ being inscribed in the strictly larger set $A$. ($A$ and $B$ have at least one boundary point in common. $B$ “fits snugly” in $A$ ...
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80 views

Perimeter points in triangle

Let $ABC$ denotes a triangle and $p(ABC)$ denotes its perimeter. We say two points $O_1$ and $O_2$ inside this triangle are perimeter points if there are points $a$, $b$ and $c$ on the sides $BC$, $AC$...
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1answer
134 views

Projection of convex set onto a convex set [closed]

Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions? Can projection of $n$ dimensional convex sets in $\mathbb R^{n'}$ ...

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