All Questions
234 questions with no upvoted or accepted answers
14
votes
0
answers
417
views
Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
- 1,902
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
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
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
11
votes
0
answers
328
views
How many ways to flatten a Tesseract onto a table?
A cube can be cut and flattened out onto a table in a way that the faces stay connected and none of them overlap. There are $384$ ways to make the cuts and $11$ distinct meshes emerge (see here). And ...
- 171
11
votes
0
answers
726
views
Making a convex polyhedron with two sheets of paper
Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that $S$...
- 4,757
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
10
votes
0
answers
333
views
Bi-spherical polyhedra
Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...
- 151k
9
votes
0
answers
1k
views
Maximum volume cross-section of a hypercube
This is surely well known, but:
Q1. What is the $(d{-}1)$-dimensional polytope
that realizes the maximum volume cross-section of a unit hypercube
by a $(d{-}1)$-dimensional hyperplane?
...
- 151k
9
votes
0
answers
543
views
Maximum volume convex body coverable by a unit square
Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...
- 151k
8
votes
0
answers
171
views
Which -icial sets produce the "standard" representations of symmetric groups?
Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...
- 18.4k
8
votes
0
answers
1k
views
Infinite Linear Programming
I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
- 133
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
1k
views
Closed-form solution of a linear programming question
Among all the probability matrices
\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \...
6
votes
0
answers
256
views
What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
- 723
6
votes
0
answers
237
views
Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
- 13.9k
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
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
82
views
What tools can show that (possibly irregular) dodecahedra do not fill space?
(Formerly on MSE.)
Here is a fairly natural question: Can three-dimensional space be filled with convex polyhedra of the same incidence structure (if not the same geometry) as the regular dodecahedron,...
- 3,068
5
votes
0
answers
162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
- 1,150
5
votes
0
answers
232
views
Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
- 18.4k
5
votes
0
answers
307
views
Which (polytopal) fans/polytopes are secondary?
Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...
- 902
5
votes
0
answers
167
views
A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
5
votes
0
answers
194
views
A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
5
votes
0
answers
204
views
A polytope associated with the Hadamard Transform
In an investigation of whether or not a subset $V$ of "Hamming Space" $M_n = \mathbb{F}_2^n$ is a tile (i.e. whether $M_n$ can be written as a disjoint partition of translates of $V$) in http://arxiv....
- 4,636
5
votes
0
answers
581
views
When is polytope compatible with network flow?
A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
- 3,475
4
votes
0
answers
46
views
Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
4
votes
0
answers
82
views
Classification of space-filling (by identical copies) convex polyhedra in R^3
Is the classification of space-filling (by identical copies) convex polyhedra in R^3 is known ?
There are only 5 "parallelohedra" - filling by translation.
But if relax that property to ...
- 24.9k
4
votes
0
answers
147
views
Name for facet of a cone containing all but one edge
Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...
- 27.8k
4
votes
0
answers
202
views
$\ell^1$-norm minimization duality
I am looking for an explicit description and discussion of the dual of the $\ell^1$-norm minimization problem $\lVert A x\rVert_1\to\min$, where $A$ is a matrix, and $x$ belongs to the $n$-simplex $\...
- 17k
4
votes
0
answers
539
views
Using Linear Programming as an iterative procedure
Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint....
- 391
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
4
votes
0
answers
229
views
Harborth's conjecture and polyhedra
Harborth's conjecture states that every planar graph can be drawn on a plane only using staight line segments of rational or integral edge length.
( There is a good mathoverflow page for this ...
- 481
4
votes
0
answers
2k
views
Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes
One can intersect a dodecahedron with a plane and
obtain an equilateral triangle, a square, a regular pentagon,
a regular hexagon, and a regular decagon:
&...
- 151k
4
votes
0
answers
790
views
Is it possible to use linear programming to solve this problem?
I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...
- 41
3
votes
0
answers
105
views
Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
- 231
3
votes
0
answers
53
views
Endpoints of intrinsic diameter of a convex polyhedron
Let $P$ be a convex polyhedron in $\mathbb{R}^3$, and $d(P)$ its intrinsic diameter,
i.e., the longest shortest surface path between two points. Say that $P$ is of
class
$D_0$ if neither endpoint of $...
- 151k
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781
3
votes
0
answers
111
views
"Slim" directed polytopes: any established name for them?
This is a "looking for context" question.
Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume ...
- 765
3
votes
0
answers
87
views
Additional symmetries of the Traveling Salesman Polytope
Given the complete graph $K_n=(V,E)$, the Traveling Salesman Polytope is a convex polytope in $\Bbb R^E$ obtained as the convex hull of the indicator vectors of (edge-sets of) Hamiltonian cycles in $...
- 13.6k
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
3
votes
0
answers
115
views
Pyramids whose volume can be computed by simple cutting and glueing
Since this question remained without answers even after a bounty, I thought it might be time to ask it here.
For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...
- 4,422
3
votes
0
answers
86
views
An exponential integral over a closed convex polytope
For any $T\geq 2$, let us define the polyhedron $S$ given by
\begin{align*}
S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\}
...
3
votes
0
answers
163
views
A new "adversarial" Wasserstein distance?
Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...
- 555
3
votes
0
answers
178
views
Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
- 131
3
votes
0
answers
193
views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
user138661
3
votes
0
answers
142
views
Can bellows make loops?
Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
- 4,600
3
votes
0
answers
222
views
Reconstructing plane graphs from degree- and face-sequences
Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...
- 151k
3
votes
0
answers
315
views
Factorization of tropical polynomials
I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here a tropical polynomial ...
- 2,246