Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
692 questions
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Isometric embeddings of $c_0$ into metric spaces
Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
- 1,151
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691
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Invariants of a set of real unit vectors in 3d space, under SO(3)
I have a set of $n$ real unit vectors, in 3-dimensional space.
(It is a follow up of Sets of vectors related by a rotation.)
Is there a construction providing a complete set of independent*) ...
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132
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If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry
Let $(X, d)$ be a compact metric space.
We say that $\{x_1, \cdots, x_n\} \subseteq X$ is an $\varepsilon$-covering of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, ...
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A geometric proof that there are infinitely many primes?
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, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
- 3,064
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87
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Instances of c-concavity outside of optimal transport?
Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
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5
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813
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Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
- 309
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95
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Effective radius of section of a convex set compared to that of the convex itself
The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai
Volumes of Sets of Constant Width in High Dimensions)
to be the radius of the $n$-ball that has ...
- 469
3
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1
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249
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Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$
A simple polygon in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by the list of the coordinates of its vertices. I expect there might be something interesting to ...
- 613
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418
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Generalization of Tucker circle, Conway circle and van Lamoen circle
Theorem 9.1 in this paper as follows is a generalization of Turker circle. Turker circles is a generalization of many circles as: Cosine Circle, circum circle, First Lemoine Circle, Gallatly Circle, ...
- 1,776
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415
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Length spectrum and Zoll surfaces of revolution
The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...
- 151k
3
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2
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185
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Lattice-point-free buffers around circles
Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
&...
- 151k
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146
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Chord of fixed length traveling around a Jordan curve
Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the ...
- 1,763
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151
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Concavity of distance to the boundary of Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ ...
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An new equilateral triangle related to the Morley triangle
Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...
- 1,776
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238
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Move one element of finite set out from A in plane
Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
- 18.8k
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109
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What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?
The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
- 24.9k
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Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
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517
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Threading pinholes in the wall of cylinder to pass through an internal coordinate
Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...
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147
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Understanding why $\frac{\phi^5}{2}$ solves this 3D optimization problem, where $\phi$ is the golden ratio
I would like to understand the deep meaning of a solution which arises from an optimization problem discussed in a paper of mine since it can be simply stated as $\frac{\phi^5}{2}$, where $\phi := \...
- 1,451
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118
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Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
3
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1
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197
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Three-dimensional Apollonian spirals
Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let $P_{\...
- 19.7k
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1
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115
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Is the max-centre map continuous for open bounded domains?
Let $A$ be an open bounded subset of euclidean $n$-space $\mathbb{R}^n$. For $x\in A$, let $r=r(x)$ be the maximal radius such that the ball centred at $x$ with radius $r=r(x)$ is contained in $A$, i....
- 2,274
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1
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107
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To find the convex planar region minimizing diameter when area and perimeter are given
The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
- 5,979
2
votes
1
answer
256
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Equidistant points on a compact Riemannian manifold
Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
- 356
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1
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101
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Another lemma on intersections of $d$-simplices
Let $d\ge1$. A $d$-simplex $S$ is the convex hull in $\mathbb R^d$ of the vertices $v_0,\dots,v_d\in\mathbb R^d$ where $\{v_1-v_0,\dots,v_d-v_0\}$ is a linearly independent set of $d$ vectors; for ...
- 1,644
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94
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Dodecahedron deformation II
(Follow-up to this question)
Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
- 2,773
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1
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300
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If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 1,149
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302
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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 ...
- 5,979
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248
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On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
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Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?
Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e.,
$N_\varepsilon^o (E)$ is the smallest number of open balls whose radii ...
- 835
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95
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Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?
I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice.
I know that it is exist for an square lattice https://mathworld.wolfram.com/...
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124
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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 ...
- 5,470
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On cutting convex regions with average values of quantities minimized
This post continues from Cutting convex regions into equal diameter and equal least width pieces - 2 and Cutting convex regions into equal diameter and equal least width pieces - 3
A basic (and to my ...
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1
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84
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'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
- 5,979
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159
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Do all compact manifolds admit geodesic tiling
Let $M$ be a compact Riemannian manifold. I'll call a set of non-empty subsets $C_1,\dots,C_N$ a geodesic tiling of $M$ if:
Each $C_n$ is closed (geodesically) convex hull of a finite number of $\{...
- 5,405
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1
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484
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Determining A valid Convex Hexagon given The length of Six sides
Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?
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646
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Mean -> Frechet mean, Standard deviation ->?
Given a finite set $A$ of points of a metric space $(X, d)$, I would like to
find its mean. A Frechet mean seems appropriate here: $\arg \min_{x \in X} \sum_{a \in A} d(x, a)^2$. I also would like ...
2
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0
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81
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Euler line in metric space [closed]
If the answer about my question here is "yes" Coplanar set in metric space.
We shall have the concept of four coplanar points in metric space $(\Bbb M,d).$ I propose an idea about Euler line in ...
- 705
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2
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720
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Euclidean triangulation of the plane with degree 7 at each vertex.
Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even ...
- 1,101
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232
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Is an orthogonal projection of a Lipschitz domain still a Lipschitz domain?
Let $\mathcal{X}\subseteq\mathbf{R}^n$ be a Lipschitz domain, i.e., for each $x\in\partial\mathcal{X}$, there exists a radius $r_x>0$ and a Lipschitz continuous function $F^x:\mathbf{R}^{n-1}\to\...
- 21
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1
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504
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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
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331
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what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
2
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2
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168
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Can all countable $CAT(0)$ cube complexes be isometrically embedded in $l^1(\mathbb{N},\mathbb{R})$?
In this paper (theorem 2), Chepoi & Hagen say
There exists an infinite $CAT(0)$ cube complex $X$ with constant maximum degree which cannot be isometrically embedded into a Cartesian product of ...
- 152
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2
answers
578
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Axiomatizing closeness: the reciprocal triangle inequality
Is there a chance to make a sound argument for the triangle inequality - characterizing distances - from general considerations only, e.g. like this:
Given arbitrary distances $d(x,y)$ and $d(y,z)$ ...
- 9,720
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1
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130
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Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
- 12.5k
2
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1
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198
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Question $B_5 \equiv B_1$ or $B_5 \ne B_1$?
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1, A_2, A_3, A_4$ be four arbitrary points lie on $(C_1)$; $B_1$ be arbitrary point on $(C_2)$. The ...
- 477
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742
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Inspired from British Flag theorem
I get this from inspired from British Flag theorem
British Flag theorem: Let $P$ be a point in the plane, let $ABCD$ be a rectangle in the plane then:
$$PA^2+PC^2=PB^2+PD^2$$
The theorem holds if P ...
- 1,044
2
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1
answer
1k
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Minimizing the Perimeter of a polyomino
Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...
- 23
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0
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278
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A chain of six circles associated with six points on a circle (in Mobius plane) [closed]
I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).
This is a generalization of the last my previous question in Three chains of six circles. (Noting ...
- 1,044
2
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1
answer
1k
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Doubling metrics, doubling measures, Lebesgue density
As stated in this question,
Lebesgue differentiation theorem holds on locally doubling space?
and proved here,
http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf
the Lebesgue differentiation theorem (...
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