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

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346 views

Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and noncompact Calabi-Yau manifolds. Nuclear Physics ...
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73 views

Gromov-Hausdorff distance measure between minimum spanning trees

I am trying to compare minimum spanning trees through time. I have two questions: 1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
3
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1answer
81 views

Gauss-Bonnet formula for 2-dimensional Alexandrov spaces

EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov. Questions 1. Can one define a measure $K$ on $S$ (thought ...
11
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2answers
296 views

Double kissing problem

Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls? An easy ...
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1answer
56 views

Convex hulls have longer boundaries

Let $C$ be a rectifiable simple closed curve in $\mathbb{R}^2$ and let $D$ be the boundary of the convex hull of the region bounded by $C$. What is the most efficient way to prove that $D$ is ...
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2answers
125 views

Differentiability of polytope shadow areas

Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$, and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$. For a point $x$ on $S$, let $\sigma(x)$ ...
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1answer
114 views

Best polygonal approximation to a polynomial $\pm$ c

Let a planar region $R$ be defined by the vertical range bounded by a polynomial $f(x) \pm c$ with $c>0$ a constant, and with $x$ varying between the smallest and largest roots of $f(x)$. For ...
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4answers
711 views

How much redundancy resides in an $n \times n$ orthogonal matrix?

Suppose one has an $n \times n$ orthogonal matrix $M$: $$ \left( \begin{array}{ccc} 0.239326 & 0.846726 & 0.475161 \\ 0.768893 & 0.13356 & -0.625272 \\ 0.592897 & ...
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5answers
307 views

Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem: Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that form an obtuse angle with one another. This was proved1 as a corollary of a lemma about ...
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2answers
122 views

Geodesics on convex hypersufaces

Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior. Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest ...
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2answers
132 views

Alexandrov spaces which are not limits of Riemannian manifolds

Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with ...
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0answers
116 views

when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...
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1answer
115 views

Regions on a sphere that avoid a fixed point set

Let $P$ be a finite set of points on a unit-radius sphere $S$ in $\mathbb{R}^3$. Treat $P$ as a fixed pattern that can be rigidly slid around $S$ as a unit (no reflection). Let $R$ be a subset of ...
4
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0answers
76 views

Relation between Euclidean distance matrices and squared-distance matrices

Let us refer to $D$ as the matrix with $D_{ij} = ||x_i - x_j||_2$ and $H = D \circ D$ e.g. the matrix with $H_{ij} = ||x_i - x_j||_2^2$ as its entries. Are the spectra of these matrices related at ...
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7answers
1k views

What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$. $P$ could be a polyhedron, or a smooth shape, or an arbitrary shape; I'll assume below that $P$ is a (non-degenerate, perhaps ...
6
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2answers
221 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 $2n$ ...
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0answers
59 views

Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial ...
11
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2answers
313 views

The most number of points that realize only $k$ distinct distances

For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$ in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances, $\|p_i-p_j\|$. Example. For points in the plane ...
20
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2answers
651 views

Is every closed curve in 3D a geodesic on a genus-0 surface?

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$. Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a ...
2
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0answers
57 views

Euclidean minimum spanning trees intersecting each unit square

The recent question "Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell" can be restated in terms of "minimum spanning trees intersecting each (closed) lattice square of an ...
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1answer
42 views

Algorithm to find the vertices of the equidistant lines between N closed polygonal lines

I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by ...
2
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0answers
132 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$. Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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0answers
124 views

Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell

Given an $n \times n$ grid with unit grid cells, and one point from the interior of each cell, what is are best possible lower and upper bounds for lengths of minimum spanning trees? The lower bound ...
5
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1answer
115 views

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 ...
4
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1answer
156 views

Another 2D theorem which generalizes to higher dimension

I have found a new property of the Tetrahedron. In fact, this is a 3D generalization of theorem 1 in my paper "A Note on Reflection", published by Forum Geometricorum. Consider any Tetrahedron ABCD. ...
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1answer
109 views

Extreme points of convex hull of Minkowski sum [closed]

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_n\}$. Let $P = \operatorname{conv}(a_1,\ldots,a_p)$ and $Q = \operatorname{conv}(b_1,\ldots,b_q)$ be two convex sets ...
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3answers
183 views

Isometric imbedding of finite metric space into standards spaces [duplicate]

Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space? (For $n=3$ this is true.) If not, what are ...
23
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1answer
451 views

Expected number of vertices of a hypercube slice — is this new/interesting?

I am a (mostly) amateur mathematician, but my education and work have featured a lot of mathematics, and recently I bumped into a mathematical problem for which I can find no references, and I am ...
16
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1answer
370 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
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1answer
142 views

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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0answers
41 views

Projection of a ray onto a random polytope

Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by ...
4
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2answers
262 views

Breaking a rectangle into smaller rectangles with small diagonals

Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
6
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2answers
240 views

A (possibly boring) Voronoi Game

The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$. Below I illustrate with $\cal C$ an equilateral triangle. Two players, $A$ and $B$, alternate turns. At each turn they ...
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2answers
264 views

What are the applications of Voronoi diagrams in pure mathematics? [closed]

Voronoi diagrams have interesting mathematical properties and applications in algorithms and modeling. But what are its applications in pure mathematics? For example, what theorems can be proved using ...
6
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1answer
187 views

Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) , I started wondering if the center of gravity is always contained in the closed convex hull. More precisely, given ...
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0answers
41 views

Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
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87 views

What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a product of trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ ...
11
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2answers
297 views

Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
5
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1answer
256 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
8
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1answer
323 views

Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...
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1answer
139 views

Can every large point set be connected to a given knot?

Let $K$ be a given knot, and $P$ a set of points in $\mathbb{R}^3$ in general position, general position in the sense that no three points are collinear and no four coplanar. Define the point-set ...
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2answers
192 views

approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers: Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...
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1answer
69 views

Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$

The largest (by edge length) regular simplex inscribed in a unit cube is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:     Image sources: left: NMSU, right: Mathworld. A recent ...
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2answers
54 views

Volume of a region given by a Constraint Satisfaction Problem

I have a Linear Constraint Satisfaction Problem i.e. I have variables $ x_1, x_2,...,x_m$, with corresponding domains $D_1,D_2,...,D_m $ satisfying linear constraints $C_1, C_2,...,C_n$ with $n ...
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2answers
300 views

A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
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0answers
29 views

Biggest volume parallelotope inside the union of two parallelotopes

Given a parallelotope $P$ symmetric around the origin, and a vector $v$, such that $(P+v)∩(P−v)$ is not empty, is there a simple way to obtain a parallelotope $Q⊂(P+v)∪(P−v)$, symmetric around the ...
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2answers
179 views

Biggest parallelogram inside the union of two translated parallelograms

If I have a parallelogram $P$ symmetric around the origin, and a vector $v$, such that $(P+v)\cap (P-v)$ is not empty, is there a simple way to obtain the parallelogram $Q\subset (P+v) \cup (P-v)$, ...
1
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1answer
75 views

Alexandrov spaces of constant curvature

Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
4
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3answers
352 views

Must the powers of some element always grow linearly with respect to a word metric?

Suppose we have a group $G$ which is finitely generated , and let $|\cdot |$ denote some word metric on it. Must there be an element $a\in G$ such that $|a^n|\ge c\cdot n$ for some $c>0$? My ...
9
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1answer
135 views

bi-Lipschitz gluing

Let $H$ be the Heisenberg group with left invariant sub-Riemannian metric and $\varepsilon>0$ is small. Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$. I have a bi-Lipschitz ...