All Questions
Tagged with mg.metric-geometry discrete-geometry
188 questions with no upvoted or accepted answers
26
votes
0
answers
359
views
Can 4-space be partitioned into Klein bottles?
It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles,
or into disjoint unit circles, or into congruent copies of a real-analytic curve
(Is it possible to partition $\mathbb R^3$ ...
- 151k
21
votes
0
answers
735
views
+300
Snakes on a plane
A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...
- 20.5k
20
votes
0
answers
433
views
Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
- 13.6k
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
17
votes
0
answers
731
views
Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
16
votes
0
answers
298
views
Realization spaces of 3-dimensional polytopes with fixed face areas
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
- 31.2k
14
votes
0
answers
270
views
Regular $n$-gon with diagonals: bounds on area of largest cell?
Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).
I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
- 3,527
14
votes
0
answers
479
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?
After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
- 532
13
votes
0
answers
378
views
Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
13
votes
0
answers
573
views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
- 3,068
10
votes
0
answers
177
views
Minimum reflection paths in a mirror polygon
Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles,
and is non-self-intersecting;
also known as a rectilinear polygon.
Treat every edge of $P$ as a ...
- 151k
10
votes
0
answers
493
views
Rectangology and squareology
I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this question ...
- 3,549
10
votes
0
answers
1k
views
Interpolating points with minimum curvature constraint
I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...
- 151k
9
votes
0
answers
144
views
Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
9
votes
0
answers
187
views
Cubing the cube - as 'perfectly' as possible
Ref: https://en.wikipedia.org/wiki/Squaring_the_square
A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...
- 5,979
9
votes
0
answers
100
views
A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
- 13.6k
9
votes
0
answers
237
views
Herding sheep in a polygon
Imagine sheep fill a simple (simply connected) polygon $P$, except
at one vertex $x$ there is no sheep.
One convex vertex $g$ of $P$ is a gate through which the sheep should pass.
A herding dog sits ...
- 151k
8
votes
0
answers
149
views
Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
8
votes
0
answers
123
views
Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?
A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...
- 13.6k
8
votes
0
answers
249
views
Approximating a general rectangle partition by a guillotine partition
There is a rectangle $R$ partitioned into some axes-parallel rectangles:
The goal is to construct another partition of $R$ into rectangles, using only guillotine cuts.
That is: cut $R$ into two ...
- 3,549
8
votes
0
answers
544
views
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
Let R be a region on a hypersphere. Each point A of the hypersphere
is associated with a vector pointing to A and with origin at
the centre of the hypersphere. So let me identify each point with a
...
- 1,207
8
votes
0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
7
votes
0
answers
316
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
7
votes
0
answers
162
views
Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
7
votes
0
answers
227
views
Tiling space with supertile of hypercube unfoldings
Two students in my class
asked and answered what might be a novel question.
It is well known that the cube has exactly $11$ edge-unfoldings
(or "nets"), as shown below:
(Image from ...
- 151k
7
votes
0
answers
254
views
Set of unit vectors such that among any three there is an orthogonal pair
I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as:
Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
- 105k
7
votes
0
answers
201
views
Minimizing energy on $\mathbb{S}^2$ for absolutely monotonic type potentials
For potential functions $f:[-1,1]\rightarrow \mathbb{R}$, satisfying that $f^{(k)}(t)\geq 0$, for $t\in(-1,1)$ and all $0\leq k \leq m$, and $f^{(m+1)}(t)<0$ for $t\in(-1,1)$, is it true that a ...
- 3,209
7
votes
0
answers
122
views
Discrepancy of the finite approximation of the Lebesgue measure
Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
- 109k
7
votes
0
answers
209
views
Stabbing disks in space, or: Galactic alignment
I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...
- 151k
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
6
votes
0
answers
74
views
Roundest polyhedra: how well can we bound the edge count of their faces?
By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
- 3,612
6
votes
0
answers
219
views
How big a box can you wrap with a given polygon?
Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of ...
- 5,979
6
votes
0
answers
112
views
Which $n$-gons of diameter 1 maximize the moment of inertia?
Background: Among convex plane $n$-gons of unit diameter, we can try to achieve:
the largest area. (This is called the biggest little polygon with $n$ sides; for $n$ odd, the regular polygon on $n$ ...
- 5,979
6
votes
0
answers
247
views
An extension of Erdos' distinct distances problem based on circles of various radii
Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the ...
- 24.7k
6
votes
0
answers
1k
views
How to pack 27 $a\times b\times c$ blocks into a cube of side $a+b+c$ with some kind of symmetry?
Recently I stumbled on the problem quoted here about a geometric proof of the AM-GM inequality $$(a_1+\cdots+a_n)^n\ge n^n a_1\cdots a_n$$ by packing $n^n$ rectangular $ n$-dimensional boxes of sides $...
- 13.4k
6
votes
0
answers
164
views
Sets of points avoiding small angles
(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...
- 151k
6
votes
0
answers
191
views
Cut locus on a hypercube
Inspired by the question, "Shortest path connecting two opposite points on a cube":
Q. What does the cut locus with respect to one corner of a hypercube
in $\mathbb{R}^d$ look like?
"The cut ...
- 151k
6
votes
0
answers
114
views
Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces
Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
- 161
5
votes
0
answers
177
views
Tiling with triangles of same circumradius and inradius
Consider a pair of positive real numbers $r$ and $R$ with $r<R/2$. Then we can form infinitely many triangles all with circumradius $R$ and inradius $r$.
For any such pair, the resulting triangles ...
- 5,979
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
93
views
Which polytopes can be deformed while keeping their edge-lengths?
Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while
keeping its combinatorial type, and
keeping its ...
- 13.6k
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
139
views
On convex regions containing (and contained within) a given triangle
Given an arbitrary triangle T.
How does one find the convex region C_M of largest area containing T such that T is also the largest area triangle that is contained within C_M?
Guess: for any T, ...
- 5,979
5
votes
0
answers
508
views
Longest simple path through hypercube corners
This is a variation on a previously answered question,
Longest path through hypercube corners.
Here I am seeking the longest simple (non-self-intersecting) path through
the unit hypercube's vertices,
...
- 151k
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
5
votes
0
answers
311
views
Biggest (or large) rectangle in a polytope
I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
5
votes
0
answers
214
views
Visibility in a prime orchard
This suggests a variant on Polya's orchard problem.
That problem asks1
for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...
- 151k
5
votes
0
answers
1k
views
N-balls covering n-balls
This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
- 151