**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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 ...

**2**

votes

**1**answer

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 ...

**5**

votes

**2**answers

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 $\sigma(x)$ ...

**4**

votes

**1**answer

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 ...

**18**

votes

**4**answers

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 & ...

**7**

votes

**5**answers

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 ...

**6**

votes

**2**answers

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 ...

**6**

votes

**2**answers

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 ...

**3**

votes

**0**answers

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 ...

**6**

votes

**1**answer

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**

votes

**0**answers

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 ...

**21**

votes

**7**answers

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**

votes

**2**answers

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 $2n$ ...

**5**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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**

votes

**0**answers

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 $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...

**5**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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. ...

**1**

vote

**1**answer

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 ...

**1**

vote

**3**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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**

votes

**2**answers

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**

votes

**2**answers

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 ...

**4**

votes

**2**answers

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 ...

**8**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**4**

votes

**1**answer

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 ...

**7**

votes

**2**answers

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 ...

**4**

votes

**1**answer

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 ...

**2**

votes

**2**answers

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 ...

**11**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**5**

votes

**2**answers

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**

vote

**1**answer

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**

votes

**3**answers

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**

votes

**1**answer

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 ...