Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,406 questions
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How "should" I define "absolutely continuous" functions on e.g. n-spheres?
(Am writing this post in a rush, out of office, so cannot give adequate links etc right now.)
There is a classical and well-understood definition of what it means for a continuous function $f:[a,b]\...
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393
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Recovering a polyhedron from its tumble-density profile
Imagine a white convex polyhedron $P$ tumbling randomly about its fixed center of gravity (c.g.)
$c$ against a blue background.
A long-exposure photo would show pure white in a neighborhood of $c$
(...
- 151k
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3
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Metric spaces as algebraic systems
Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, ...
user6976
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Reference Request: Perspective Painting
What is a good book/article explaining the mathematics behind perspective painting? I have already looked at the Wikipedia article on the topic, so I am looking for something more advanced than this. ...
- 3,284
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2
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Problem equivalent to "largest square in a cube"
The "largest square in a cube" problem, which asks for the largest square inside a cube, has a solution as can be seen on this page, which also says that the general problem in higher dimensions is ...
- 7,320
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347
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Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
- 93
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181
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Expected value of the length of the shortest non-zero vector in a lattice?
$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
- 609
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430
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Geometric proof of the three-dimensional Pythagorean theorem
All the proofs of the high-dimensional Pythagorean theorem that I know are based on induction or the additivity of the dot product. Is there any geometric construction that's similar to the well-known ...
- 1,431
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284
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Extending a partially defined metric on a metrizable space
Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}_+$ a continuous function that satisfies the ...
- 278
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237
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m-point-homogeneous, but not (m+1)-point-homogeneous
It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ ...
- 45k
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2
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377
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Convex surfaces with minimal total curvature in Cartan-Hadamard 3-space
A Cartan-Hadamard 3-space $M$ is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature. A (smooth) convex surface $\Gamma\subset M$ is an embedded ...
- 7,179
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577
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Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines
The simplest case of the Fundamental Theorem of Projective Geometry states that, if $f: \mathbb{R}^2\to\mathbb{R}^2$ is a bijection that preserves lines – in the sense that if $L\subseteq\mathbb{R}^2$ ...
- 2,896
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218
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Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense
Let $\{X_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X_i$ is ...
- 21.8k
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185
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Cohn-Vossen rigidity theorem in hyperbolic space
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...
- 21.8k
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Volume doubling, uniform Poincaré, counterexample
The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
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212
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Geometry of complements to compacts of codimension 2
Let $K\subset \mathbb{R}^n$ be a (nonempty) compact of covering dimension $\le n-2$. In particular, $K$ does not separate $\mathbb{R}^n$ (even locally). I will equip $M=\mathbb{R}^n-K$ with the ...
- 31.2k
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168
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Are $\varepsilon$-connected components dense?
Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of ...
- 42k
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281
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Convex sets in Alexandrov spaces
Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary?
Definition of a convex subset: $A\subseteq X$ ...
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132
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Can we reconstruct the region in the xy plane by length measurements?
Consider a closed smooth bounded curve enclosing a region $S$ in the XY-plane $\mathbb{R} ^2$.
We define the function $f(x)$, where $x$ is a point on the $x$ axis, as the length of the intersection ...
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314
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Are rays in Carnot groups straight?
A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...
- 3,146
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2
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207
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Volume satisfying inequality constraints (simplex subset)
Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...
- 61
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410
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Existence of finite set of points in the revolving circles
Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...
- 4,784
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718
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What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?
Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
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330
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Best and worst centrally symmetric convex covering shapes
Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw
down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$,
until $S$ is ...
- 151k
6
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868
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Shortest geodesic loop vs. shortest periodic geodesic
Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...
- 13.5k
6
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591
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For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
user14166
6
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2
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217
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Untangling entwined rigid chains in 3-space
I am interested in exploring the degree of "tangledness"
of two rigid chains in space.
A polygonal chain is a simple (non-self-intersecting) path
of segments in
$\mathbb{R}^3$, viewed as a rigid body. ...
- 151k
6
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524
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Enlarging a tetrahedron with integer edge lengths
Given a tetrahedron with all edges having integer length, is it always possible to increase all of the edge lengths by one?
More precisely: Let $P_1, P_2, P_3, P_4$ be four distinct non-coplanar ...
- 856
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609
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$W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere
Hi,
it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 \rbrace\hookrightarrow\...
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2
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643
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A convex polyhedral analog of the pentagram map
I am wondering if there is a three-dimensional analog of
the pentagram map, which maps a convex polygon to another
convex polygon. Here's the Wikipedia image:
I am seeking something similar that maps ...
- 151k
6
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2
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945
views
Vortex Voronoi diagram?
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and ...
- 151k
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How to define a Voronoi reduced basis?
Let $\Lambda$ be an $n$-dimensional lattice with basis $b_1,\ldots,b_n$. The problem of finding a "good" basis for $\Lambda$, or reducing a "bad" basis into a good one, is a very active area of ...
- 1,085
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2
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189
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Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π.
Given a convex polygon, how does one algorithmically find the point (...
- 5,979
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258
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Expected doubling constant of a random Erdős–Rényi graph
Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (...
- 5,405
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928
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To find the Largest Regular n-gon contained in a given convex region
Given a general convex region C, to find the largest regular polygon that is contained in it (shared boundaries allowed). Basically, one needs to find that particular value of n for which a regular n-...
- 5,979
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261
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Area of $n$-sphere contained outside $\ell_1$ ball
For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...
6
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232
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Pascal's theorem for spherical hexagon
I draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case.
My question 1. Is there simple proof for this?
My question 2. Can we change the circle on sphere ...
- 705
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336
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Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis
Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$.
It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...
- 419
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560
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Volume ratio of general $\ell_p$ balls and surfaces
This question is a generalization of the question Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces
For any $p\in[1,\infty]$ define $\|x\|_p := (|x_1|^p+\cdots+|x_d|^p)^{1/p}$ for $p\in[1,\infty)$ ...
- 373
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Current interest in geometric properties of Hilbert fundamental domains
Harvey Cohn published several articles in the 1960's analyzing geometric properties of fundamental domains for Hilbert modular surfaces.
H. Cohn, "On the shape of the fundamental domain of the ...
- 2,436
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236
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Pigeonholing Polygons: Can two rigid regions fit in twice the space needed?
This is a tweak of Henry Segerman's question
Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1? , but restricted to the point of possibly having a
proof in the ...
- 2,158
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205
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Hiding $k$ disks inside a larger disk
Suppose one has $k$ unit-radius disks, and the goal is to hide them inside
a disk of radius $R \gg k$.
The detection probes are rays along a line.
(Think of the disks as tumor cells, and the rays as ...
- 151k
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185
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Maximizing ratio volume/diameter^n by an affinity
Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.
It is clear that there is a constant $\...
- 5,063
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886
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theorems equivalent to the parallel postulate
Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?
- 19.7k
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525
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Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity
Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...
- 7,197
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426
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A toy model for the t-section problem
Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H $ is some prescribed function (small, smooth, vanishing near the endpoints to any ...
- 61.9k
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803
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Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 143
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260
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Arbitrary-dimensional expanders?
Rephrasing expansion (slightly). Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph.
(We write a weighted graph as $(V,\beta)$, where the weight $\...
- 20.2k
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388
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Covering number estimates on closed Riemannian manifolds
Let $(M^n,g)$ be an $n$-dimensional compact and connected Riemannian manifold with sectional curvature bounded above and below by $c,C$. Is it possible/known how to express the external covering ...
- 195
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180
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Seeking a Weyl tube formula for Whitney stratified spaces
Background: Let $X$ be a smooth, compact Riemannian submanifold of euclidean space $\mathbb{R}^n$. H Weyl's tube formula asserts that for sufficiently small $t > 0$, the volume $V(X;t)$ of the ...
- 15.5k