All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
1
vote
1
answer
96
views
A 'natural' enumerable metric space with integral distances which is essentially the Euclidean space
It is easy to construct a metric space $E_d$ such that all points
of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space ...
- 17.9k
1
vote
1
answer
89
views
Vertices of 2 self-polar triangles lie on conic
I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?
- 49
1
vote
1
answer
228
views
Lattice points in hypercubes
Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
- 503
1
vote
1
answer
226
views
Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier
Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...
- 851
1
vote
0
answers
68
views
Name of the perspector of the orthic triangle and excentral triangle
The orthic triangle and tangential triangles of a given triangle are in perspective. What's the official kimberling center associated with this perspector?
- 1,109
1
vote
0
answers
67
views
Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
1
vote
0
answers
42
views
On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
1
vote
0
answers
59
views
What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
- 5,979
1
vote
0
answers
44
views
On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
- 5,979
1
vote
0
answers
52
views
'Self-similar and perfect' partitions of planar regions
Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition.
A classical example ...
- 5,979
1
vote
0
answers
100
views
Perfect 'cuboiding' of cubes and cuboids
We try to add a bit to ref 2 listed below. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special ...
- 5,979
1
vote
0
answers
40
views
Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
- 5,979
1
vote
0
answers
91
views
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....
- 5,979
1
vote
0
answers
109
views
Which polygons allow partition into rational triangles?
A triangle with all side lengths rational is said to be a rational triangle. It is known - for example, Cutting the unit square into pieces with rational length sides - that the unit square allows ...
- 5,979
1
vote
0
answers
53
views
The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
- 5,979
1
vote
0
answers
72
views
Intercept theorem in $\mathbb R^n$
The celebrated intercept theorem(also known as Thales's theorem) provides the ratios between the line segments created when two parallel lines are intercepted by two intersecting lines.
I'm looking ...
- 193
1
vote
0
answers
52
views
On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
- 5,979
1
vote
0
answers
57
views
Inside-out dissections of solids
We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...
- 5,979
1
vote
0
answers
41
views
About the number of faces of the conification of a polytope
Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
- 131
1
vote
0
answers
69
views
Least number of squares of size N that a set of R rectangles can occupy
Given a set $R$ of rectangles of different positive integer sizes, and any number of squares of the same size $N\in\mathbb{N}$, what's the least number of squares $C$ that all the rectangles together ...
- 119
1
vote
0
answers
46
views
Kissing behavior of planar regions
This post reworks a question that was stated in a slightly different form at Convex region $C$ with least kissing number of copies of $C$.
Background: Given a 2D region $C$ (not necessarily convex), ...
- 5,979
1
vote
0
answers
38
views
Possible extensions of the perpendicular axes theorem for moment of inertia
This post continues on Moment of inertia from Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia.
The perpendicular axis theorem states that the moment ...
- 5,979
1
vote
0
answers
77
views
Lattice packing
Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...
- 237
1
vote
0
answers
100
views
All 3-dimensional symmetric reflexive polytopes
$\DeclareMathOperator\Conv{Conv}$I am finding all 3-dimensional symmetric reflexive polytopes. To do so, first, we know that all 2 dim symmetric reflexive polytopes are $X_3=\Conv((-1,-1),(1,0),(0,1))$...
- 21
1
vote
0
answers
48
views
Inside-out dissection
In a recent problem in The College Math Journal (1230) a Heronian triangle is called to have an equivalent rectangle if there exists an integer sided rectangle with the same area and perimeter. For ...
- 111
1
vote
0
answers
85
views
More on triangles inscribed in convex regions with one vertex fixed
We add a bit to On maximum perimeter triangles inscribed in convex regions with one vertex fixed. Let C be a convex planar region and P a point on its boundary.
Are there convex shapes C other than (...
- 5,979
1
vote
0
answers
111
views
Maximizing the minimum curvature of a convex shape with a given volume in higher dimensions
Given any $d$-dimensional convex shape $S$ in the Euclidean space with $d\gg 1$, let $K_{\min}(S)$ be the minimum value of the Gaussian curvature of its boundary.
Question: What is the maximum value $...
- 1,677
1
vote
0
answers
40
views
Tiling with a one-parameter family of non-congruent triangles
This post continues Tiling with triangles of same circumradius and inradius.
The following are known about infinite sets of triangles that can be parametrized with one variable:
from an infinite set ...
- 5,979
1
vote
0
answers
76
views
Convex planar regions with optimal average 'centralness' and 'depth'
For a planar convex region $C$ and an interior point $P$ we define:
the centralness ratio at $P$ is
$$\min\left(\frac{\text{shorter portion of }\chi}{\text{longer portion of }\chi}:\chi\text{ is a ...
- 5,979
1
vote
0
answers
56
views
Are sharper lower bounds known for these potentials on the sphere?
Fix a positive integer $\ell$. For $x_1,\dotsc,x_n\in S^{d-1}$, Venkov proved that
$$
\sum_{i=1}^n\sum_{j=1}^n(x_i\cdot x_j)^{2\ell}\geq\frac{(2\ell-1)!!(d-2)!!}{(d+2\ell-2)!!}\cdot n^2,
$$
with ...
- 7,633
1
vote
0
answers
153
views
Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
- 550
1
vote
0
answers
57
views
Shadows and planar sections of polyhedra – 2
This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies
Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal ...
- 5,979
1
vote
0
answers
68
views
Facility location and traveling salesman
This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
- 5,979
1
vote
0
answers
124
views
A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
- 5,979
1
vote
0
answers
46
views
Multi-layered wrapping of polyhedra
This post continues from How big a box can you wrap with a given polygon? and Convex polyhedra that can be folded from convex polygons. One can also mention 'k-fold coverings of the plane' as examined ...
- 5,979
1
vote
0
answers
48
views
Deployment and dispersion in triangular regions
Definitions (from C. Stanley Ogilvy's 'Tomorrow's Math'):
Deployment: To place a specified number $n$ of points (stations) in a region such that the maximum distance of any point in the region from ...
- 5,979
1
vote
0
answers
124
views
Number of lattice points in a structural symmetric convex body
Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space
\begin{equation}
\...
- 550
1
vote
0
answers
81
views
Constructive way to optimally cover a compact subset of Euclidean space
Let, $(X,d)$ be a simply connected compact subset of $\mathbb{R}^d$ with non-empty interiorn, let $d$ denote the Euclidean metric, and let $\varepsilon>0$. Is there a way to iteratively select ...
- 5,405
1
vote
0
answers
45
views
How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
- 13.6k
1
vote
0
answers
64
views
Euclidean embedding of the mesh
$M$ is a topological mesh, i.e. triple $M=(V,E,F)$, where $V$ is the vertex, $E$ is the edge and $F$ is the face, such that $M$ is homeomorphic to the sphere.
Suppose that we have a metric $l :E\...
- 1,915
1
vote
0
answers
69
views
Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?
General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
- 13.6k
1
vote
0
answers
196
views
Squares as sum of squares
For which positive integers n is $n^2$ the sum of precisely n smaller positive squares?
Of these n x n squares, which can be actually cut into n smaller squares?
- 3,707
1
vote
0
answers
70
views
Covering number of the range of a function
I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
- 183
1
vote
0
answers
80
views
Euclidean embedding of a graph based on 1-ring neighborhood distances only
Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...
- 337
1
vote
0
answers
60
views
Finding special vectors generated by a matrix
Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
- 13.9k
1
vote
0
answers
57
views
Covering the annulus of symmetric convex body
Consider a symmetric convex body $A$ in $\mathbb{R}^d$. Now, we draw another object, $A'$, concentric and translated with respect to $A$ and having radius slightly greater than twice to the radius of ...
- 285
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
1
vote
0
answers
450
views
When is the conical hull of a finite set of vectors a subset of the space? (and tilings)
Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...
- 1,314
0
votes
1
answer
229
views
Is this bounded?
May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...
- 1