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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More on triangles inscribed in convex regions with one vertex fixed
We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.
Are there convex shapes C other than (...
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If all the chocolate is within distance r of the outer boundary of the choco egg, what is the max.quantity of chocolate contained within a unit ball?
We have proved the following statement, but wonder if this result is actually known (reference??)
It solves the following problem. Suppose you have a (possibly) hollow chocolate egg whose outer ...
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142
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A question about Roger Penrose's spin networks and mathematical formalization?
Let $a,b,c$ be "units" in the spin network.
Then there are there are the following three requirements to fulfill (according to the relevant Wikipedia entry):
$a,b,c \in \mathbb{N}$
Triangle ...
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A question on a paper of B. S. Henriksen
I have been reading the article "A peak set of Hausdorff dimension $2n-1$ for the algebra $A(\mathcal{D})$ in the boundary of a domain $\mathcal{D}$ with $C^\infty$-boundary in $\mathbb{C}^n$&...
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Is the standard $\mathbb R^4$ the only one with positive sectional curvature?
Perelman--Cheeger--Gromoll Soul Theorem states that whenever a complete non-compact Riemannian manifold $(M,g)$ has positive sectional curvature, it should be diffeomorphic to an Euclidean Space.
On ...
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Barnes-Wall lattices’ contact polytopes
The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
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Construction of a homogeneous Moran set
Fix a positive integer $N\ge 2$, for $n \in \mathbb{N}$, denote
$$\Sigma=\{0,1,\dots,N-1\},\\
\Sigma^n=\{(\omega_1,\dots,\omega_n):\omega\in\Sigma, i=1,\dots,n\}.$$
Let $p>2$ be a positive integer. ...
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A question about angles in an irregular tetrahedron
Given an irregular tetrahedron $ABCD$ with a circumscribed sphere.
This defines $4$ spherical triangles.
For the vertex $D$ these triangels are $ABD$, $BCD$ and $ACD$.
How to prove that
$$
\cos^2 ADB +...
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Maximizing the minimum curvature of a convex shape with a given volume in higher dimensions
Given any $d$-dimensional convex shape $S$ in the Euclidean space with $d\gg 1$, let $K_{\min}(S)$ be the minimum value of the Gaussian curvature of its boundary.
Question: What is the maximum value $...
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Maximum number of concurrencies for $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$
I have $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$ and want to prove that there cannot be more than $L$ points where $J$ hyperplanes intersect simultaneously (aka. concurrencies).
I suspect that the ...
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$L^1$-valued Lipschitz extension problem on a simplex
Consider a regular $n$-simplex, and a map from the vertices to $L^1$.
How can we find the minimum Lipschitz constant of an extension of this map to the entire simplex?
Is there any literature or ...
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A claim of Gromov in "Metric Structures for Riemannian and non-Riemannian Spaces" [closed]
I'm trying to understand an example that occurs very early in Gromov's book "Metric Structures for Riemannian and Non-Riemannian Spaces".
On page 3, he introduces the following metric on $\...
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A metric geometry problem which calculates the limitation of human eyes
This is the update version of this question A functional inequality which calculates the limitation of human eyes
Let an Euclidean space $M$ (or a path connected metric space) be partitioned into ...
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Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius.
The following are known about infinite sets of triangles that can be parametrized with one variable:
from an infinite set ...
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Example of CAT($k$) space [closed]
Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
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Question about coarse fixed point property in large-scale geometry
I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...
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449
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Bound on covering number of Lipschitz functions – missing part in proofs of Kolmogorov et al
Given a metric space $(\mathcal{X},\rho)$ and $\mathcal{A}\subset\mathcal{X}$ totally bounded, i.e. $\mathcal{A}$ has a finite $\varepsilon$-covering for any $\varepsilon>0$. Consider $\...
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Volume of a polytope as its degenerates to be lower dimensional
Consider a polytope $P$ defined by the usual inequalities $A\mathbf{x}\leq \mathbf{b}$; let me assume that $P$ is not contained in a proper subspace. A result which I believe to true, but am not ...
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Measure estimates of $\delta$-neighbourhood of compact sets
I am interested in the estimating from above the measure of a compact set $K$ by a sequence of sets $K_n$, converging to it in the Hausdorff metric. As such I am looking for known conditions that give ...
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Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$
Let $|\cdot|$ denote the usual Euclidean norm on $\mathbb{C}^3$ and fix some arbitrary metric $\rho$ on $\mathbb{CP}^2$. For $\delta > 0$ and any set $\hat{P} \subset \mathbb{CP}^2$, define the $\...
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Are Carnot groups ever CAT(𝜅) spaces?
Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?
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Maximizing a parametric integral over the unit sphere
I am trying to compute the nonnegative quantity
$$
\underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty
$$
where $\...
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Any round convex body between a simplex and a ball?
Fritz John's ellipsoid theorem gives the minimal ball containing a given convex body $K$. Moreover, we have for $m$ points in boundary
$$\tag{$A$}\label{A}\sum_m c_iu_iu_i^{T} = I_n,$$
where $I_n$ is ...
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Are cells of 4-polytopes a convex polyhedron by definition?
I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...
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117
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
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How to eliminate angle in a Glissette equation of carried point of a line sliding along two lines not at right angles
Glissettes are the curves traced out by a point carried by a curve, which is made to slide between given points or curves. My problem specifically include a line which slides between two fixed lines (...
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Angles between edges of a geometric graph and graph invariants
Are there any clever ways in which the angles between edges in a geometric graph are encoded in the graph spectrum, or another object associated with the graph?
I'm interested to see what else is ...
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Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
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Doubly ruled surfaces in hyperbolic 3-space
A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...
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Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube
This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
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Is any geometry also a Klein geometry?
Can any metrical or axiomatic geometry be shown to be isomorphic to a Klein geometry?
Can any manifold with a (pseudo) Riemannian metric be defined in the Kleinian sense?
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Best estimate on doubling constant of a finite metric space
Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant?
Probability based on its cardinality, diameter, and ...
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Convex planar regions with optimal average 'centralness' and 'depth'
For a planar convex region $C$ and an interior point $P$ we define:
the centralness ratio at $P$ is
$$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\text{ is a ...
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Efficient algorithm for a distance on strings
Let $(M,d)$ be a metric space. Consider two sequences $a = (a_i)_{i=1}^n$, $b = (b_i)_{i=1}^m$, $n, m \in \mathbb{N}$ with elements in $M$. For two sequences $[n],[m]$, call $$\Gamma([n], [m]) = \big\{...
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Are sharper lower bounds known for these potentials on the sphere?
Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2,
$$
with ...
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Influence of the degenerate Delaunay tiles on the Voronoï diagram
About three or four years ago, I implemented the Delaunay and Voronoi tessellations in Haskell, with the help of the Qhull C library. Now I reimplement it in R.
I have noticed that including or not ...
- 2,560
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153
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Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
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Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
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82
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Intersecting lattices with surfaces in R^d
Let us fix some bounded surface $S\subset \mathbb{R}^d$. Let $x_1,\ldots, x_m$ be some non-zero vectors in $\mathbb{R}^d$. I am interested is the maximum number of points that the lattice $L_m=\{\sum ...
- 2,288
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To extend the Steiner-Lehmus theorem
The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles.
Question: What could one say ...
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Constant width curves and inscribed/ circumscribed ellipses
It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
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Short lattice vectors in the complement of a hyperplane
Suppose that $\Lambda \subseteq \mathbb{R}^n$ is a lattice and $H \subseteq \mathbb{R}^n$ is a hyperplane such that $H \cap \Lambda$ has rank $n - 1$. I would like to know an upper bound on the ...
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Why Densest packing of equal spheres in three dimensions is not 88.86? [closed]
I placed four spheres of radius R at vertices of a tetrahedron of edge length 2R .When I calculated density I got 88.86.Actualy I wanted to calculate what is the maximum number of earth that can be ...
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Pseudo-Droz-Farny circles
I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
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Shadows and planar sections of polyhedra – 2
This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies
Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal ...
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Facility location and traveling salesman
This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
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Reference to equivariant Gromov-Hausdorff convergence
I am looking for a reference to the following notions and facts below which, I think, I can prove, but which might be known to experts.
Let us fix a finite group $G$. Consider the class of all compact ...
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71
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Diminishing of the $4_{21}$
One of the projections of the $4_{21}$ polytope (https://en.m.wikipedia.org/wiki/4_21_polytope) into four dimensions positions its vertices as those of two concentric 600-cells scaled by the golden ...
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Injectivity of post-composition operator
Let $X$, $Y_1,Y_2$, and $Z$ be separable metric spaces. Let $C(X,Y)$ be the topological space of continuous functions from $X$ to $Y$ equipped with its compact-open topologies. Fix a continuous ...
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A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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