Questions tagged [mg.metric-geometry]

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

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Do triple-linked graphs exist?

Lets say that a finite simple graph $G$ is (intrinsically) fully triple-linked if for each embedding of $G$ into $\Bbb R^3$ we can find three disjoint cycles $C_1,C_2,C_3\subset G$ whose embeddings ...
M. Winter's user avatar
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2 votes
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Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $X\to\ell^\infty$ given by taking $x_n$ the countable dense subset and sending $x\mapsto\lvert(x,x_n)-d(x,x_0)\rvert$. This ...
Gesh's user avatar
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Three annulus intersection problem [migrated]

Recently I faced a problem about intersection of three annuli. Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles....
Stefan Vujic's user avatar
1 vote
1 answer
105 views

Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary. Given $f\colon M\to N$ be an $\varepsilon$-...
asv's user avatar
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Algorithm to generate configurations with kissing number 12

That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
GRquanti's user avatar
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Does there exist a plane curve such that it has the heart curve as catacaustic?

Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$. The catacaustic ...
zemora's user avatar
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Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
RandomTensor's user avatar
1 vote
2 answers
256 views

Does $\mathbb Z^n$ contain $A_n$?

Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
WKC's user avatar
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Seek a partition of $\mathbb R^d$

Let $c>0$ be given. I look for $n\ge 1$ and a collection of closed subsets $(F_i: 1\le i\le n)$ such that $$\bigcup_{1\le i\le n} {\rm int}(F_i)= \mathbb R^d,$$ and for every $x\in \mathbb R^d$, ...
GJC20's user avatar
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To place copies of a planar convex region such that number of 'contacts' among them is maximized

A contact between two planar convex regions obviously happens either along a line segment or at a single point. Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
Nandakumar R's user avatar
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Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
mathoverflowUser's user avatar
1 vote
1 answer
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When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
asv's user avatar
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3 votes
1 answer
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"Almost geodesics" in Riemannian manifolds which cannot be loops

Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
asv's user avatar
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6 votes
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When is a distance space dominated by a metric space?

A distance space is a pair $(X,d)$ where $X$ is a set and $d:X \times X \rightarrow \mathbb{R}$ is a symmetric, non-negative map such that $d(x,x)=0$ for all $x \in X$. These are sometimes called semi-...
David Bryant's user avatar
1 vote
1 answer
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To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region

We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above ...
Nandakumar R's user avatar
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3 votes
1 answer
75 views

Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$

Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition: Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
Daniel Asimov's user avatar
2 votes
0 answers
72 views

Nested convex hulls in Hadamard manifold

Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood. Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$? ...
Anton Petrunin's user avatar
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What is the dual of a hyperbolic configuration of points?

Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{...
Malkoun's user avatar
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A claim on the largest area circular segment that can be drawn inside a planar convex region

This post adds a little to To find the longest circular arc that can lie inside a given convex polygon A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
Nandakumar R's user avatar
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3 votes
0 answers
83 views

Sampling uniformly from the convex cone

Let $n$ vectors of dimension $d$ (e.g., $n = 100$, $d = 10000$), each with infinity norm of $1$, be given. The conic combination of those $n$ vectors generates a convex cone. How to uniformly sample ...
mathhamcs's user avatar
3 votes
0 answers
145 views

What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
Gro-Tsen's user avatar
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1 vote
2 answers
183 views

A triangle comparison in CAT(0) spaces

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
Mohammad Ghomi's user avatar
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45 views

Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?

By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C. Question: For specified ...
Nandakumar R's user avatar
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2 votes
1 answer
162 views

Forming paper bags that can 'trap' 3D regions of max surface area

An existence question based on 'Trapping' 3D regions with sheets of paper. Given a sheet of paper S that is a planar convex region, one tries to form a 'closed bag' that contains a connected ...
Nandakumar R's user avatar
  • 5,371
4 votes
1 answer
126 views

Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$. Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
Dmitrii Korshunov's user avatar
2 votes
0 answers
36 views

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 \...
Brendan Mallery's user avatar
2 votes
1 answer
149 views

Osculating sphere at point of maximal curvature lies to one side

I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
JMK's user avatar
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0 answers
68 views

On 'Width Equalizers' of planar convex regions

Definitions: The least width of a 2D convex region C is the least distance between any pair of parallel lines that both touch the boundary of C (in what follows, we refer to this quantity as simply '...
Nandakumar R's user avatar
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3 votes
1 answer
146 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
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7 votes
2 answers
279 views

Étendue measure of the set of lines between two Euclidean balls

Let $d>0$ and $r_1,r_2>0$ such that $r_1+r_2 < d$. Consider two (say, closed) balls $B_1,B_2$ in $\mathbb{R}^m$ having radii $r_1,r_2$ and whose centers are at distance $d$. Let $C$ be the ...
Gro-Tsen's user avatar
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1 vote
0 answers
79 views

$L^p$-compression of metabelian groups

Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
ARG's user avatar
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3 votes
0 answers
107 views

Reference request: Carathéodory-type theorem for convex hulls of closed sets

I'm looking for a reference for the following theorem. Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
Tom Leinster's user avatar
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1 vote
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Can the volume of a neighborhood of the cut locus be arbitrarily small?

Let $(M^n,g)$ be a complete, $n$-dimensional Riemanian manifold without boundary, maybe non-compact. Let $p\in M$ be a point, and $C_p$ the cut locus. It's known that $C_p$ has Hausdorff dimension $\...
mathmetricgeometry's user avatar
5 votes
0 answers
73 views

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 $H(f)$ ...
Dmitrii Korshunov's user avatar
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0 answers
57 views

Comparing partitions of a given planar convex region into pieces with equal perimeter and pieces of equal width

We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
Nandakumar R's user avatar
  • 5,371
3 votes
0 answers
84 views

Minkowski problem for polytopes: the origin of necessary condition

Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets. Theorem (Minkowski). Let $A_i$ be positive faces areas ...
Alexey Ustinov's user avatar
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0 answers
89 views

Bounding the area of the image of a set by product of maximum of lengths

Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$. My question feels ...
JustSomeGuy's user avatar
2 votes
0 answers
69 views

Converse of existence of minimizers

Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...
Rohan Didmishe's user avatar
3 votes
2 answers
753 views

Kepler conjecture: Are there only two most efficient packings or could there be more than two?

Today I attended a talk by Terence Tao, attended by (I'm guessing) probably at least a couple of thousand people, in which among other things he said it had been proved that no packing of spheres in ...
Michael Hardy's user avatar
2 votes
1 answer
92 views

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 ...
Tri's user avatar
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0 votes
0 answers
33 views

Determining a convex hyperbolic pentagon by all side lengths and two specified angle sums

We are trying to prove the following statement for convex hyperbolic pentagons which we believe should be true. Consider a convex hyperbolic pentagon with sides of lengths $a, b, c, d, e$. Suppose the ...
Shiv Parsad's user avatar
1 vote
0 answers
42 views

Computational complexity of exact computation of the doubling dimension

Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
98 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
259 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
Alainty's user avatar
  • 11
1 vote
0 answers
35 views

Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap

Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
MMH's user avatar
  • 111
11 votes
0 answers
473 views

Are there 100 points that are part of every half-density part of the plane?

Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$? I am deliberately being vague ...
domotorp's user avatar
  • 18.2k
2 votes
0 answers
136 views

$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$

Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
No-one's user avatar
  • 1,035
0 votes
0 answers
28 views

Minimizing the number of grid squares to cover a polygon

Given an arbitrary polygon, and a grid square size x, I'd want to find a placement of the polygon such that it covers the minimum amount of cells in the grid. The ...
b9s's user avatar
  • 101
0 votes
1 answer
52 views

Robustness of doubling dimension to small perturbations

Let $M$ be a metric space. Then the doubling dimension of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls ...
pyridoxal_trigeminus's user avatar
0 votes
1 answer
72 views

Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
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