All Questions
109 questions
5
votes
2
answers
304
views
Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?
The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...
- 13.6k
5
votes
1
answer
303
views
Intersection of rotating regular polygons
This question has a recreational flavor, but may not be
entirely uninteresting.
Let $P_k$ be a unit-radius regular polygon of $k$ sides,
and $P_n$ a unit-radius regular polygon of $n \ge k$ sides.
...
- 151k
5
votes
1
answer
176
views
Orientations of triples of points in the plane
Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements ...
- 19.7k
5
votes
1
answer
361
views
What is known about the duals of cyclic polytopes?
What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...
- 13.6k
5
votes
1
answer
397
views
How much of an aperiodic tiling is needed to force aperiodicity?
Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
- 20.2k
5
votes
1
answer
114
views
Packing in uniform domains
Given $N$ points $X:=(x_i)_{i \in \{1,..,N\}}$, we now define a score function $S:X \rightarrow \mathbb{N}$ that is $S(X)= \sum_{i=1}^N S(x_i)$ where the score of $S(x_i)$ is
$$S(x_i) = 2* \vert \{x_j;...
- 536
5
votes
0
answers
235
views
Arrangement of points, lines, and planes
Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties?
every line is incident with four points and ...
- 2,773
5
votes
0
answers
190
views
The existence of $n$-sided cells in regular $m$-gons
For any integer $n >= 3$, does there exist a regular
$m$-gon with all diagonals drawn containing a cell with $n$ sides?
See A342222 and its cross-references.
Regular polygon on the Wiki.
&...
- 251
5
votes
0
answers
313
views
Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes
Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement?
There exists ...
- 1,677
5
votes
0
answers
199
views
Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size
Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$.
Let $s(C)$ and $\ell(C)$ be ...
- 1,677
5
votes
0
answers
135
views
What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?
Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...
- 13.4k
4
votes
3
answers
347
views
Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
- 13.6k
4
votes
1
answer
298
views
Does Kalai's $3^d$ conjecture hold for simplicial spheres?
Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...
- 13.6k
4
votes
2
answers
94
views
Finding a not too slim triangulation with prescribed vertices on $\mathbb R^2$
Let us fix a constant $r>1$. Let $d(x,y)$ denote the distance between points $x,y\in \mathbb R^2$. Suppose we have a discreet subset $X\subset \mathbb R^2$ such that
1) For any two points $x,x'\...
- 14.3k
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
4
votes
1
answer
422
views
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
For $k = 2$ the answer is obvious since we can always place circles so that every one of them ...
- 63
4
votes
2
answers
287
views
Problems similar to Borsuk’s Theorem in the plane
Consider a 2-dimensional Borsuk's theorem:
Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$.
I wonder if there are any results ...
- 181
4
votes
1
answer
293
views
Number of points in a lattice and an oblong box
I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
- 20.2k
4
votes
1
answer
491
views
Generalization of the "double cap conjecture" to a vector space with complex field
The conjecture that I proposed in
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun.
See for ...
- 1,207
4
votes
0
answers
66
views
Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 364
4
votes
0
answers
222
views
What does it mean "parallel"?
I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following
Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear ...
- 41.9k
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
0
answers
246
views
Distance properties of the permutations of a set of points in a Euclidean space
We are given a set of $n$ distinct points $S=\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}$ in a Euclidean space $\mathbb{R}^d$, where the distance between two points $\mathbf{x}_i,\mathbf{x}_j\...
- 1,677
4
votes
0
answers
81
views
Number of orders of distances between points on a line
Points $a_1, a_2, \dots, a_n$ on a line form a set from $n(n-1)/2$ distances between them. Suppose all that distances are different, numerating them from the shortest to the longest one we obtain some ...
- 1,431
4
votes
0
answers
94
views
Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius
I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
- 161
4
votes
0
answers
443
views
Intersection of pencils in $\mathcal{R}^2$
Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...
3
votes
1
answer
222
views
Number of lines of symmetry of a set of lattice points
Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of $\...
- 1,890
3
votes
2
answers
438
views
If a polytope is centrally symmetric and combinatorially equivalent to a zonotope, is it a zonotope?
A zonotope is a polytope whose 2-faces are centrally symmetric.
Question: If a polytope $P$ is centrally symmetric and combinatorially equivalent to a zonotope, is it itself a zonotope?
- 13.6k
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
3
votes
1
answer
218
views
Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions
We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
- 1,677
3
votes
1
answer
143
views
Combinatorial Euclidean geometry problem
Let $\mathcal{S}^d_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given ...
- 1,677
3
votes
2
answers
344
views
Is a vertex- and edge-transitive polytope already a uniform polytope?
I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...
- 13.6k
3
votes
1
answer
190
views
How many points are in such set with the same norm-2
Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...
3
votes
0
answers
187
views
Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
- 1,677
3
votes
0
answers
134
views
Two questions on counterexamples to Borsuk's conjecture and ball-packings
In 1933 Karol Borsuk conjectured the following
Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$?
Whilst new to this ...
- 31
3
votes
1
answer
484
views
On some infinite planar arrangements with triangles
Background: Given a convex region C. One can define a graph corresponding to a planar arrangement of non overlapping congruent copies of C - each unit C is a node and an edge connects it to another ...
- 5,979
3
votes
0
answers
137
views
Aperiodic tile with rational area
Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
- 745
2
votes
1
answer
143
views
Triangles and convex hulls in high dimensions
Given a set $S_n$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, such that every $(d+1)$-tuple in $S_n$ is affinely independent, and let $C(S_n)$ be the convex hull ...
- 1,677
2
votes
1
answer
370
views
Large subgroups of the Hamming cube
Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube).
Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
2
votes
1
answer
151
views
Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance
We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
- 1,677
2
votes
2
answers
255
views
What is the smallest number of subsets in such a subdivision?
Given any $30$ points in the plane, what is the smallest number of
subsets in a subdivision of the set of $30$ points into subsets such
that all the points in each subset are on the boundary of the ...
- 21
2
votes
1
answer
404
views
Euclidean distance bound with geometric constraints
Let $S_n$ be a set of $n$ points belonging to $\mathcal{B}_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|_2\le 1\}$, where $d\ll \log(n)$.
Let $s_n$ and $\ell_n$ be respectively defined as follows:
$$...
- 1,677
2
votes
1
answer
153
views
Bounding number of $k$-nearest neighbor sets in $\mathbb{R}^d$
Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ is compact.
Let there be $n$ distinct points $X = \{ x_1,...,x_n \} \subseteq \mathcal{X}$ and $k = \lfloor n^\alpha \rfloor$ where $0 < \alpha &...
- 23
2
votes
1
answer
84
views
What is the average component size of a coloring?
Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
- 13.4k
2
votes
1
answer
151
views
Graph immersed into the plane with segments as edges and we search for matching with no edges intersecting
There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph immersed into the plane where the points are the vertices and the ...
- 286
2
votes
1
answer
247
views
Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?
This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
2
votes
0
answers
131
views
Maximum number of regions in a disk partitioned by pairs of parallel chords
We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,...
- 1,677
2
votes
0
answers
131
views
Optimal way to group points in the plane into clusters
Consider a strictly decreasing sequence $d = (d_k)_{k\ge 1}$ of distances in $(0,1)$. Given a constant $C>2$, we say that $d$ has the $C$-grouping property if any finite non-empty subset $S$ (of ...
- 1,761
1
vote
2
answers
227
views
A question about dense sets
Suppose that $A$ is a given subset of $I=[0,1],\ $ and
$ \left\{ x_j = \frac{j}{m} \right\}_{j=0}^{m}\ $ is the $m$-partition of $I$, and $\nu(m)$ is the number of
$\ [x_{i-1},x_{i}]\ $ such that $\ [...
- 419
1
vote
1
answer
221
views
What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
- 5,405