All Questions
65 questions
2
votes
2
answers
163
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References for geometric properties of optimal Euclidean traveling salesman tour
Consider a finite set of points $V \subseteq \mathbb{R}^2 $ as a TSP-instance under the standard $\| \cdot \|_2$ norm. (TSP stands for traveling salesman tour.) We know that every optimal TSP tour $T$ ...
CommunityBot
- 1
3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
22
votes
2
answers
900
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...
- 32.1k
3
votes
0
answers
93
views
Minkowski problem for polytopes: the origin of necessary condition
Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets.
Theorem (Minkowski). Let $A_i$ be positive faces areas ...
- 12.3k
1
vote
2
answers
232
views
What does the extension theorem for tilings state?
I have seen several references to the so-called Extension Theorem in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads
The Extension Theorem [......
- 35.5k
3
votes
0
answers
135
views
Intersecting the unit n-cube and (n-1)-planes
(Is this a known problem?)
Question Let $\ 1<n\in\mathbb N.\ $ What is the greatest $(n-1)$-area
$\ S(n)\ $ of $\ L\cap I^n\ $ where $\ I^n\subseteq\mathbb R^n\ $ is the unit cube, and $\ L\ $ ...
- 6,337
2
votes
0
answers
233
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
21
votes
2
answers
1k
views
Forbidden mirror sequences
Let $\cal{M}$ be a finite collection of two-sided mirrors,
each an open unit-length segment in $\mathbb{R^2}$,
and such that the segments when closed are disjoint.
A ray of light that reflects off the ...
- 2,821
16
votes
4
answers
2k
views
Point sets in Euclidean space with a small number of distinct distances
It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
- 2,821
34
votes
6
answers
8k
views
Covering a unit ball with balls half the radius
This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks":
How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of ...
- 2,821
6
votes
2
answers
544
views
On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
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21
votes
5
answers
1k
views
Is a rhombus rigid on a sphere or torus? And generalizations
If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...
- 4,713
11
votes
2
answers
1k
views
Which (semi)regular polyhedra are combinations of two others?
The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and $...
- 4,713
5
votes
1
answer
266
views
Contracting a set to a ball
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$
Question 1: Let $S$ be a nonempty measurable subset of $\R^n$. Let $B$ be a closed ball in $\R^n$ such that $m(B)=m(S)$, where $m$ is the Lebesgue ...
- 128k
2
votes
2
answers
164
views
Angle between a point in a convex polytope and the nearest point of a face
Let $P \subset \mathbb{R}^d$ be a convex polytope, and let $F$ be a face of $P$ (of co-dimension 1, let's say). Now let $x \in P \setminus F$ and let $y \in F$ be the nearest point of $F$ to $x$. Then ...
- 11
26
votes
7
answers
3k
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 non-...
- 1,446
15
votes
2
answers
737
views
Tiling survey that updates "Tilings and patterns"?
Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.
I am ...
- 63.9k
9
votes
3
answers
1k
views
Generalization of Sylvester-Gallai theorem
The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
of points so that a line through every two of them passes through a
third unless they are all on a single ...
- 32.1k
5
votes
2
answers
307
views
Tiling a Jordan polygon
I saw this problem some years ago, don't remember the source:
Let $P$ be a Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) that can be tiled with ...
- 18.6k
15
votes
2
answers
863
views
Three squares in a rectangle
One of my colleagues gave me the following problem about 15 years ago:
Given three squares inside a 1 by 2 rectangle, with no two squares overlapping, prove that the sum of side lengths is at most 2. (...
CommunityBot
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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 ...
- 151k
14
votes
3
answers
2k
views
Optimal wireframe sphere
Suppose you have a length $L$ of metal pipe at your disposal,
and you would like to build a wireframe unit-radius sphere,
by bending, cutting, and welding the pipe into a connected structure $F$.
Your ...
- 63.9k
3
votes
1
answer
381
views
Source on counting lattice points on a line
Looking for a book or article on the result linked below. The result tells us that the number of lattice points on a line between points $(a,b)$ and $(c,d)$ is given by $\gcd(a-c,b-d)+1$.
https://math....
- 198
5
votes
1
answer
190
views
Finding a superbase in a lattice of Voronoi first kind
An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that
$\{b_1,\ldots,b_n \}$ is a ...
- 2,183
3
votes
1
answer
111
views
Reference for "every 5-dimensional polytope has a 3-gonal or 4-gonal face"
It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...
- 35.5k
14
votes
2
answers
878
views
Sets of evenly distributed points in the Euclidean plane
Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection
with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?
If the answer is yes, can $P$...
- 19k
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
6
votes
4
answers
2k
views
Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
- 151k
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 ...
- 4,713
13
votes
1
answer
3k
views
What nets fold to polyhedra?
There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
CommunityBot
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17
votes
5
answers
883
views
Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed
Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges ...
- 14.3k
11
votes
1
answer
406
views
Thinnest 2-fold coverings of the plane by congruent convex shapes
It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...
- 151k
21
votes
2
answers
1k
views
On convergence of convex bodies
Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\...
- 7,194
45
votes
1
answer
2k
views
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
- 151k
7
votes
3
answers
805
views
Wrapping a convex polyhedron with string
This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P \...
- 18.6k
14
votes
1
answer
819
views
The geometry of crinkled aluminum foil
I wonder if the geometry of crinkled aluminum foil has been studied?
The above is a photo of foil I flattened to reuse.
It might be ...
- 151k
17
votes
1
answer
458
views
The sparsest planar net that captures every unit segment
Let $\cal C = \lbrace C_i \rbrace$ be a collection
of rectifiable curves in the plane with the property that
every unit-length segment meets at least one curve
in at least one point.
Call such a ...
- 151k
8
votes
1
answer
885
views
Maximal tetrahedra inscribed in ellipsoid
Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...
- 151k
4
votes
0
answers
153
views
Perimeters of nested convex spherical polygons
I seek a reference—not a proof—that if $P_1$ and $P_2$
are two convex polygons on a sphere composed of geodesic segments,
contained in a hemisphere, and
$P_1 \subseteq P_2$, then the ...
- 151k
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
4
votes
1
answer
323
views
What properties does generalized Delaunay triangulation have?
Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
- 4,713
13
votes
2
answers
1k
views
Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
- 13.6k
10
votes
2
answers
280
views
Monochromatic point sets in two-colored plane
Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
Property M (for Monochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic ...
- 151k
5
votes
2
answers
441
views
Touching-tetrahedra graphs
Have the graphs representable by touching tetrahedra been explored?
Let $\cal T$ be a collection of tetrahedra in $\mathbb{R}^3$
with pairwise disjoint interiors.
Define a graph $G_{\cal T}$ to have ...
- 151k
7
votes
1
answer
938
views
Which knots' stick numbers are twice their crossing numbers?
Looking at a table of minimum stick numbers for knots (table here),
it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$
is realized by the trefoil $3_1$—it ...
- 4,713
13
votes
1
answer
430
views
Detecting a hidden convex body with line probes
Imagine that, somewhere inside an origin-centered, unit-radius sphere
$S$ in $\mathbb{R}^3$,
sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$,
with $\alpha < 1$ the fraction of ...
- 151k
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
3
votes
1
answer
292
views
Existence of Simple Closed Straightest Geodesics
There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...
- 4,713
10
votes
2
answers
930
views
What is determined by the combinatorics of the shadows of a convex polyhedron?
Define the shadow of a convex polyhedron $P$ in direction $u$
to be the orthogonal projection of $P$ onto a plane whose normal is $u$.
The shadow is a convex $k$-gon.
I am wondering to what degree $P$ ...
- 151k
6
votes
1
answer
276
views
Matching on sphere to create cycle with chords
Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like to ...
- 151k