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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A question on Cheeger-Colding theory
I'm reading Compactification of certain Kähler manifolds with nonnegative Ricci curvature by Gang Liu recently. And I feel hard to understand a statement in the paper. Now the assumption is $(M,g)$ is ...
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An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
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Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space).
We need to ...
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Generalization of some plane geometry theorems
Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\...
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Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
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If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
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Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B ...
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Nagel line of a tetrahedron?
It's well known that there is an analogy for the Euler line in a tetrahedron, but is there also an analogy for the nagel line of a tetrahedron? I can't seem to find any decent literature talking about ...
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Surfaces with curvature $\leq K$ are of bounded integral curvature
One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$
is the ...
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Cusp points in Alexandrov spaces
Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
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Finding lattice with short basis-vectors containing given lattice
While working on understanding the space spanned by certain integer relations of real numbers I have come across the following problem. Given $v_1,\dots, v_n \in \mathbb{Z}^m$, I am would like to find ...
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Practical applications of dandelin spheres
I know that dandelin spheres can be used to prove the focal properties of conic sections, but I heard that they can be used to help track the orbits of planets. All the sources I looked up only said ...
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On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
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What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
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Are two convex solids with all corresponding shadows equal in area congruent?
By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
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Volumes of sets of constant width in high dimensions
Background
The $n$-dimensional Euclidean ball of radius $1/2$ has width $1$ in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between ...
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On the history of cone-3-manifolds
A cone-3-manifold (of constant curvature) is a geometric 3-manifold locally modelled either on the Euclidean/hyperbolic/spherical 3-space or on the respective metric cones over spherical cone-surfaces ...
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
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Is the formula known? and can we generalized for higher dimensions?
In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?
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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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Special rectangle and its existence in non-Euclidean geometries
My questions is motivated by Folding the Hyperbolic Crane article which presents non-Euclidean paper for origami and the existence of a special rectangle on Euclidean paper.
Actually, there exists a ...
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Curves on potatoes
On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler:
The puzzle is attributed to the book "The mathemagician and pied puzzler", and ...
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Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
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Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
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Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is false (...
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On maximum perimeter triangles inscribed in convex regions with one vertex fixed
Ref: Convex curves with many inscribed triangles maximizing perimeter
Given a planar convex region C. Let P be a variable point on its boundary.
Observations: When C is an ellipse, the variation in ...
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Do lattices with small covering radius have sublattices with small covering radius?
For me a lattice is a discrete subgroup of $\mathbb R^n$. The linear span of a lattice, written $\Lambda \otimes \mathbb R$, is the $\mathbb R$-vector subspace of $\mathbb R^n$ generated by $\Lambda$. ...
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Towards a metric characterization of Euclidean spaces
I want to obtain a metric characterization of the classical finite dimensional spaces of Euclidean geometry.
Motivation: Suppose $A$ and $B$ live in an $n$-dimensional Euclidean space. They are each ...
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Pushing a convex cone and equidistants
Let $K$ be a closed convex cone in an n-dimensional Euclidean space.
Suppose $K$ has non-empty interior. For $t > 0$
form the subcone $K_t$ consisting of all points in $K$ which lie a ...
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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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Rauch comparison theorem for $C^{1,1}$ metrics
If $g$ is a smooth riemannian metric on $M$ with nonpositive sectional curvature, the Rauch comparison theorem implies that $(M,d_g)$ is a negatively curved metric space (every point has a ...
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Genaralizing the metric expression present in the quadrilateral inequality
Let $(X, d)$ be a metric space. In Sato - An alternative proof of Berg and Nikolaev’s characterization of CAT(0)-spaces via quadrilateral inequality it is stated that if $X$ is a geodesic space, then ...
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Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
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A gerrymandering problem - can you always turn a tie into a landslide victory?
Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...
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Estimating direction from a distribution on a circle
Let there be $n$ points on a unit circle. It is known they come from "normal" distribution around particular unknown direction (i.e. sum of 2 "normal" distributions on circle - one centered at point $...
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
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Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
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Tracking a reference: "Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals"
I linked a paper by James Schmerl in a recent question which cites Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Privately Published, 1987.
I have had difficulty finding any ...
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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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Reference request: stability of the barycenter under the logarithm map in Hadamard spaces
Let $X$ be an Hadamard space.
For $p\in X$ let $\log_p:X\to T_pX$ be the logarithm map that maps points in $X$ to the corresponding points in the tangent space $T_pX$.
Let $μ$ be a Borel probability ...
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Product of low dimensional Hausdorff measures
Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
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On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
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Are there arbitrarily sparse "lattices" in negatively curved symmetric spaces?
Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic ...
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Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
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A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
This question was posted at MSE but was not answered.
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$...
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Progress on Gromov's Conjecture of the bound of total Betti numbers
This question is a reference request.
Let $(M,g)$ be a Riemannian manifold of dimension $n$, and $b_i(M) = \dim H_i(M,\mathbb{R})$. Gromov proved it that there are constants $C(n)$ such that, if the ...
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About planar curves on a manifold
I recently came upon the following situation (think of $\mathbb{R}^3$ to simplify): let $S$ be a compact smooth surface with $K>0$ everywhere and define
$$Q=\frac{\sup_{p}\lambda_{1}(p)}{\inf_{p}\...
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'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
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Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
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Complexity and length
Suppose we define continuous piecewise linear functions $f$ on $[0,1]$ using your favorite programming language, or by finite automata, or by any other suitable machine. Define the complexity $H(f)$ ...
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