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Questions tagged [mg.metric-geometry]

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

-2
votes
1answer
453 views

Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here: PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
3
votes
0answers
38 views

Measure of set of vectors whose outer product are bounded

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$. Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of ...
1
vote
0answers
56 views

From Sudakov minoration principle to lowerbounds on Rademacher complexity

For a compact subset $S \subset \mathbb{R}^n$ (and an implicit metric $d$ on it) and $\epsilon >0$ lets define the following $2$ standard quantities, Let ${\cal P}(\epsilon,S,d)$ be the $\epsilon-...
3
votes
1answer
98 views

Smoothness of a curve vs. smoothness of the squared distance from the curve to points on Riemann manifolds

I know that the squared distance function from a point $p$ on a Riemann manifold $M$ is smooth in a n-hood of $p$. Therefore for a smooth curve $c:\mathbb{R}\to M$ the concatenation $d(p,\cdot)^2\circ ...
2
votes
1answer
200 views

Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof: Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...
5
votes
3answers
179 views

Average caliper diameter (mean width) of a polyhedron

Define the caliper diameter of a polyhedron as follows: Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
12
votes
3answers
604 views

General principles which lead to good questions in many concrete situations [closed]

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
3
votes
1answer
133 views

Lattice projections

I imagine the following result is folklore Theorem. Those $k$-dimensional subspaces $\zeta \subset \mathbb{R}^n$ $(1 \leq k \leq n-1)$ for which the orthogonal projection of the lattice $\mathbb{Z}^n$...
1
vote
3answers
209 views

Repeatedly halve and twist a planar shape: Limiting shape?

Consider the following iterative process. Start with a planar region $R=R_0$ of $\mathbb{R}^2$. I am thinking of $R$ as connected, but it may become disconnected. In the example below, $R$ starts as ...
5
votes
0answers
75 views

What quantum groups admit quantum topography space structure?

Quantum topography space is a pair $(A,M)$ consisting of a $C^*$-algebra $A$ and an abelian sub algebra $M\subset A$ with approximate identity. The intuition is to take $M$ be the smallest abelian ...
2
votes
1answer
215 views

About Vitali covering theorem

In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderon-Zygmund ... /1998, Int. Math. Res. Not., www.math.brown.edu/~treil/papers/l1/l1-5.pdf on the ...
0
votes
1answer
190 views

Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
3
votes
1answer
115 views

Interpolation inequality related to the 5/3-Laplace operator

I'm having trouble with an estimate that would be helpful in information geometry. The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a ...
3
votes
0answers
27 views

Tangencies of Villarceau circles in a 3D Steiner chain

Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture ...
6
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0answers
137 views

Approximating a ray with an integer lattice point

Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$ With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...
3
votes
1answer
77 views

Extremum problem on regular simplex

I'm looking for a proof for the extremum problem on regular simplex. Question. Let $\mathcal{A}=A_0A_1...A_n$ be a regular simplex in $\Bbb E^n$. $P$ is a point inside and on boundary of $\mathcal{...
0
votes
0answers
55 views

Inequality on simplex with circumscribed sphere

I'm looking for a proof for this problem on simplex which I think it is true Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with ...
2
votes
2answers
117 views

Centroid and center circumscribed spheres in simplex

I'm looking for a proof for this problem on simplex which I think it is true Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center ...
2
votes
0answers
73 views

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 ...
1
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0answers
71 views

Coplanar set in metric space

Let $(\Bbb M,d)$ be a metric space. Give three points $X,$ $Y,$ $Z$ in $\Bbb M$ such that they satify one of the following conditions $i)\ d(X,Y)+d(Y,Z)=d(X,Z),$ $ii)\ d(Y,Z)+d(Z,X)=d(Y,X),$ $iii)\...
7
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0answers
175 views

A conjecture on simplex

Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$ Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
5
votes
1answer
164 views

Cyclic quadrilateral in metric space

Consider a metric space $(\Bbb M,d).$ If $X,Y,Z\in \Bbb M.$ We define cosin of angle by $$\cos(\angle YXZ)=\frac{d(X,Y)^2+d(X,Z)^2-d(Y,Z)^2}{2d(X,Y)\cdot d(X,Z)}.$$ If we have four points $A,$ $B,...
4
votes
1answer
116 views

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 ...
4
votes
1answer
132 views

Coloring circles in plane

We assume that all the circles in the plane are each colored with one of two colors: red or blue. My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...
3
votes
0answers
75 views

Miquel circles on sphere

Consider a sphere $\Bbb{S}$ on Euclide 3D space. We well-known that a "line" connecting two points $X$ and $Y$ on $\Bbb{S}$ is the great circle of $\Bbb{S}$ which passes through points $X$ and $Y.$ ...
3
votes
0answers
219 views

A generalization of the Sawayama-Thebault theorem

1. Introduction The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...
4
votes
1answer
306 views

Converse of Pythagorean theorem in n-dimensional Euclidean space [closed]

We consider a simplex $A_0A_1...A_n$ in $n$-dimensional Euclidean $\Bbb E^n.$ Denote by $Vol(A_0A_2A_3...A_n)=V_1,$ $Vol(A_0A_1A_3...A_n)=V_2,$ $...$ $Vol(A_0A_2A_3...A_{n-1})=V_n$ and $Vol(...
15
votes
2answers
399 views

A cube is placed inside another cube

I known following problem with two square $$Area(1)+Area(3)=Area(2)+Area(4).$$ My question. Is this problem true for two cubes? We place a cube $XYZT.X'Y'Z'T$ into another cube $ABCD.A'B'C'D',$ ...
1
vote
0answers
64 views

Distance between quadric surface and point or Intersection of sphere and quadric surface

I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances. Given a ...
5
votes
0answers
220 views

$N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
8
votes
1answer
228 views

Bi-Lipschitz extension

Given a bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
2
votes
0answers
108 views

Pascal theorem for three dimensions

A year ago I found the Pascal theorem for three dimentions as follows: Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
2
votes
0answers
71 views

Wasserstein distance to the set of Gaussians, relation to Boltzman dissipation rate

$\newcommand{\RR}{\mathbb{R}}$ I am interested in the 2-Wasserstein distance for probabilities over $\RR^n$, $W_2(μ,ν)=\left(\inf\int_{\RR^n×\RR^n}|w−v|^2dπ(v,w)\right)^1/2$ where the infimum is taken ...
4
votes
2answers
186 views

Fitting one Polygon in another

I have two Polygons A and B and I want to find the position, rotation and scale of B, so it fits into A and has the maximum Area possible. Also both can be concave. I did some research but couldn't ...
9
votes
1answer
333 views

Divergence of Groups and Metric Spaces

Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
14
votes
3answers
883 views

An ellipse through 12 points related to Golden ratio

I am looking for a proof of the problem as follows: Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$...
26
votes
5answers
2k views

If a triangle can be displaced without distortion, must the surface have constant curvature?

Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$). Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics. If $T$ can be moved around arbitrarily on $S$ ...
11
votes
3answers
595 views

Steiner's inequality reference request

I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\...
7
votes
1answer
167 views

Five-dimensional manifolds fibering over a fixed hyperbolic surface

I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
6
votes
2answers
174 views

Sliding through a curvature-bounded tube: Maximum volume?

My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view. Q1. Is it the case that the maximum convex volume body inside a ...
0
votes
0answers
23 views

Convex misfits, the equi-affine version

Let the input convex bodies in $\ A\subseteq \mathbb R^n\ $ have Lebesgue measure $1,\ |A|=1.\ $ For two of them, $\ A\ $ and $\ B,\ $ let $\ A+B\ $ be the Minkowski sum of them. Then $$ m(A\ B)\ =\ \...
13
votes
1answer
384 views

Regularity of geodesics

If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample. For a related discussion see Metric ...
1
vote
0answers
29 views

In 3D point groups, does $[\Gamma_{e}\otimes\Gamma_e] = \Gamma_{Rot_z} \forall$ degenerate $\Gamma_e$ hold in general?

In the following I am referring to groups exclusively describing 3D point symmetries. I use the Schönflies notation for groups and their elements and the Mulliken symbols to describe their irreducible ...
5
votes
3answers
549 views

Isometries and fixed points

I am new to geometric group theory and I am trying to read a bit to expand my horizons. I have encountered the following theorem: Suppose that $G$ is a group that has a free action by isometries on $\...
3
votes
0answers
121 views

Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures

Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
1
vote
1answer
180 views

A question about a $2^n$-point metric space

For any positive integer $n$, let $X_n$ be the family of all subsets of $\{1,2,\cdots,n\}$. Let $(X_n,d)$ be the metric space such that $$d(A,B)=|\,A\triangle B\,|,\ \forall A,B\in X_n$$ where $A\...
22
votes
1answer
422 views

Tying knots via gravity-assisted spaceship trajectories

Q. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?           To make this more ...
4
votes
2answers
737 views

What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.
2
votes
1answer
203 views

Covering number of Lipschitz functions

What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$? Only 2 results I have found so far are, That the $\infty$-...
4
votes
1answer
234 views

Reordering vertices of a polygon

Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...