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

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0
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31 views

Is the map $\exp_x(\sum_{i=1}^m d^2(x_i,x))$ Lipschitz?

The last question is too general, this is a modification. Let $M$ be an $n$ dimensional Riemannian manifold. Fix $m$ points $x_1,...,x_m$. Suppose $y$ is not in the cut locus of $x_i$ for $1 ...
11
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3answers
176 views

Curvature of a finite metric space

I am sorry to ask a very vague question, but: What are good ways to define the curvature of a finite metric space? The best way I can think of is: the curvature of a finite metric space $M$ is ...
1
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0answers
90 views

Reference request: Ebin

I'm after the paper The manifold of Riemanian metrics by D. Ebin. A link to the reference is: http://www.ams.org/mathscinet-getitem?mr=0267604 The paper seems to be very hard to track down. Can ...
53
votes
11answers
6k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
5
votes
1answer
48 views

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 ...
5
votes
2answers
763 views

Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies. Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
1
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1answer
77 views

4D Duoprisms based on nonconvex polygons

A duoprism is a polytope that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$). Four-dimensional duoprisms in particular have been studied: $$P \times Q = \{ ...
15
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1answer
474 views

Lower-Hölder embeddings of the sphere

My question is very simple: Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that $$ |f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r, $$ for ...
7
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2answers
268 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
6
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1answer
106 views

Fixed point of fatness

For the purposes of this question, define the following properties of convex sets in the plane: A set is $R$-fat (for $R\geq 1$) if it contains a disc of side-length $x$ and is contained in a disc ...
2
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2answers
69 views

Basic question about discrete minimal surfaces

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of ...
0
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0answers
50 views

Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...
1
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0answers
56 views

A compact Alexandrov space with curvature bounded below has curvature bouneded above?

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
6
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2answers
224 views

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 $2n$ ...
3
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1answer
81 views

If all ball around fised basepoints are isometric, are the spaces as well (length spaces)?

Let $X$ and $Y$ be two complete proper length spaces, $x \in X$ and $y \in Y$. Assume for every $r>0$ the closed balls $\overline{B_r(x)}$ and $\overline{B_r(y)}$ are isometric. Does there exist ...
6
votes
2answers
366 views

Geodesics on SO(3)

I have two 3D rotations about the origin, represented as $3 \times 3$ orthogonal matrices $M_1$ and $M_2$ (specified by numerical entries), and I would like to interpolate (and compute) a continuous ...
18
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5answers
850 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
2
votes
1answer
112 views

Embedding graphs into hyperbolic spaces

Do we know of a characterization as to when does a graph have a "good" embedding into a hyperbolic space? (And does having such an embedding have a spectral or wavelet analysis signature?) I don't ...
9
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3answers
2k views

Left invariant metric on SLn

I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...
5
votes
2answers
451 views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
3
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0answers
84 views

Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let $(M_i,p_i)$ be a sequence of $n$-dimensional Riemannian manifolds with lower Ricci curvature bound $-1$. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there ...
2
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1answer
69 views

Is a cocompact CAT(0) periodic?

Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...
9
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2answers
307 views

Is every metric space quasi-isometric to a graph?

I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$...during this proof I've precisely used the fact that given two point in $X$ there ...
13
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1answer
395 views

On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at ...
14
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1answer
679 views

Is it possible for a metric on a smooth manifold to be smooth?

Are there any smooth manifolds $M$ with the following property: There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$? If not, is it ...
4
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0answers
54 views

Tetrahedron incenter iteration tree

This is driven more by curiosity than by research, but nevertheless may be of some interest. Start with a regular tetrahedron $T$ with corners $(a,b,c,d)$, and let $x$ be its incenter—the ...
12
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3answers
843 views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. ...
3
votes
1answer
228 views

Maximum distance of points in intersection of balls

let $B_\delta(p):=\{x\in\mathbb{R}^d:||x-p||_2\leq \delta\}$ be a $d$-dimensional closed ball. Now I do not have one ball, but four: $B_{r_1}(p)$, $B_{r_2}(p)$, $B_{s_1}(q)$ and $B_{s_2}(q)$. ...
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2answers
237 views

Examples on small cut radius of totally convex set in non-negatively curved manifold

Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
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0answers
74 views

Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...
5
votes
3answers
233 views

set of centers of sphere inscribed in tetrahedron

Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced ...
14
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3answers
628 views

Is Monsky's theorem depending on axiom of choice ?

The extension of the 2-adic valuation to the reals used in the usual proof uses clearly AC. But is this really necessary ? After all, given a equidissection in $n$ triangles, it is finite, so it ...
7
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2answers
119 views

Kissing number and overlapping number

Let $S$ be a certain family of geometric objects (e.g, the family of unit squares). The kissing number of $S$ is the maximum number of nonoverlapping elements of $S$ that can touch one element of ...
5
votes
3answers
255 views

How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
3
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1answer
184 views

Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...
21
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2answers
1k views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
0
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0answers
47 views

Best measure for curve similarity

I would like to measure similarity between two curves represented by two arrays of points. The similarity measure should not depend on the size of these shapes. Two similar shapes but have different ...
1
vote
1answer
262 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
2
votes
1answer
87 views

How often can a single length occur as a boundary distance?

Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$? We can make any assumptions about the ...
1
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0answers
102 views

understanding geometry of eigen values of Ricci tensor [closed]

As per I can visualize the eigen value $\lambda$ of a linear map $T:V \rightarrow V$, defined by $Tv=\lambda v$, is actually the scaling factor of the vector in the same direction as of $v$.My ...
20
votes
3answers
2k views

Distributing points evenly on a sphere

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible. I have found some related questions on stackoverflow but ...
1
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1answer
66 views

Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...
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4answers
431 views

SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that any finite subgroup of SO$(3)$ (the $3 \times 3$ orthogonal matrices of determinant $1$) is either a cyclic group $C_n$, or a dihedral group $D_n$, or one of the groups ...
2
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1answer
47 views

Is there a curve on a surface where an integrable function is pointwise bounded?

I consider $\Sigma:=S^2$, as a unit radius sphere in $\mathbb R^3$ so that it has an induced metric, measure etc on it. Is the following statement true? For a large constant $K$, there exist ...
2
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0answers
202 views

Why only Normed Linear Spaces? [closed]

It is well known that "Norm on a vector space can be used to obtain a metric on that space." I think easily we can generalize the notion of norms to groups and rings. My questions are, Why ...
0
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0answers
22 views

c-superdifferential is unique +cost function is differentiable, then the potential function is differentiable?

Let $M$ be a compact Riemannian manifold, $\mu$ and $\nu$ are two Borel probability measures, the cost function $c(x,y)=\frac{d^2(x,y)}{2}$. It's well known that the infimum of the Kontorovich's ...
4
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0answers
86 views

Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
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2answers
54 views

Pairwise distance distribution for point clouds (normal distribution) [closed]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
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0answers
22 views

Concave functions on a cone over Alexandrov space

Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary. Let $X$ be the cone over $\Sigma$. $X$ is well known to be non-negatively curved. Let $f\colon X\to \mathbb{R}$ be a ...
1
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1answer
53 views

Two geodesics with angle $\pi$ in Alexandrov space

Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are not the two ...