All Questions
1,021 questions
15
votes
2
answers
1k
views
Do two new special points in any triangle exist?
There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...
- 1,776
1
vote
1
answer
123
views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
- 111
29
votes
2
answers
2k
views
Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?
In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...
- 9,720
2
votes
1
answer
273
views
Checking planar convexity of 4 points with Stewart's formula
Is the following conjecture correct?
Conjecture:
If $A,B,C,D$ are four points in general position in the euclidean plane, with
$a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
$a':=\|D-A\|,\...
- 13.2k
-1
votes
1
answer
88
views
sparse data fitting problem [closed]
I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
Given a sparse data fitting problem:
$ minimize \...
- 1
7
votes
1
answer
386
views
Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?
Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:
The all-one vector $j$ is contained in the conic hull of $col(A)$.
...
- 81
4
votes
2
answers
275
views
Hyperrectangle that contains most of cube's interior (except its vertices)
Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$.
Is there a hyperrectangle $H$ that satisfies the following:
$\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n}
\begin{cases}
[p,q], &...
- 265
3
votes
1
answer
75
views
Number of Inner Diagonals of Convex Hulls of $n+2$ Points in Convex Configuration in $E^n$
Question:
Is it true that $E^2$ is the only Euclidean space, in which the convex hull
of $n+2$ points in convex configuration has two inner diagonals and in all other cases there is only one such ...
- 13.2k
2
votes
1
answer
54
views
Triangle Center from Weighted Perfect Matchings
let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$.
Question:
what is known about the ...
- 13.2k
1
vote
0
answers
74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
12
votes
1
answer
443
views
Is each cover of the plane by lines minimizable?
A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called
$\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$;
$\bullet$ minimizable if $\...
- 41.8k
2
votes
0
answers
169
views
theories where angles exist without a metric
The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
- 2,041
9
votes
2
answers
389
views
Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?
According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen ...
- 2,130
5
votes
0
answers
119
views
What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?
Here are two facts that are superficially similar.
Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons.
Feit-Higman Theorem: The only finite ...
- 1,389
5
votes
1
answer
255
views
Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?
A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
- 41.8k
-2
votes
1
answer
587
views
Is the conjecture true for n-sphere $(n>2)$? [closed]
This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
- 1,776
5
votes
0
answers
364
views
Shapes defined by points
Can shapes determined by some number of points?
From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is dense on every closed jordan ...
- 289
7
votes
3
answers
412
views
Average caliper diameter (mean width) of a polyhedron
Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
- 101
2
votes
1
answer
179
views
Controlling angles between vectors using sum of subvector angles?
This is a technical question coming out of my research.
Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that
$$
\...
- 25
3
votes
1
answer
149
views
Tangencies of Villarceau circles in a 3D Steiner chain
Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture ...
- 2,560
4
votes
1
answer
829
views
Constructibility of the regular 17-gon [closed]
There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...
- 61
4
votes
1
answer
136
views
Extremum problem on regular simplex
I'm looking for a proof for the extremum problem on regular simplex.
Question. Let $\mathcal{A}=A_0A_1...A_n$ be a regular simplex in $\Bbb E^n$. $P$ is a point inside and on boundary of $\mathcal{...
- 705
0
votes
0
answers
136
views
Inequality on simplex with circumscribed sphere
I'm looking for a proof for this problem on simplex which I think it is true
Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with ...
- 705
4
votes
2
answers
335
views
Centroid and center circumscribed spheres in simplex
I'm looking for a proof for this problem on simplex which I think it is true
Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center ...
- 705
1
vote
0
answers
82
views
Coplanar set in metric space
Let $(\Bbb M,d)$ be a metric space.
Give three points $X,$ $Y,$ $Z$ in $\Bbb M$ such that they satify one of the following conditions
$i)\ d(X,Y)+d(Y,Z)=d(X,Z),$
$ii)\ d(Y,Z)+d(Z,X)=d(Y,X),$
$iii)\...
- 705
8
votes
0
answers
233
views
A conjecture on simplex
Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
- 705
5
votes
1
answer
194
views
Cyclic quadrilateral in metric space
Consider a metric space $(\Bbb M,d).$
If $X,Y,Z\in \Bbb M.$ We define cosin of angle by
$$\cos(\angle YXZ)=\frac{d(X,Y)^2+d(X,Z)^2-d(Y,Z)^2}{2d(X,Y)\cdot d(X,Z)}.$$
If we have four points $A,$ $B,$ $C$...
- 705
6
votes
1
answer
232
views
Pascal's theorem for spherical hexagon
I draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case.
My question 1. Is there simple proof for this?
My question 2. Can we change the circle on sphere ...
- 705
4
votes
1
answer
178
views
Coloring circles in plane
We assume that all the circles in the plane are each colored with one of two colors: red or blue.
My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...
- 705
1
vote
1
answer
129
views
Coloring lines in plane
We assume that all the lines in the plane are each colored with one of two colors: red or blue. Given angle $\alpha.$
My question 1. Is there possible to get two lines with the same color and angle ...
- 705
3
votes
0
answers
905
views
A generalization of the Sawayama-Thebault theorem
1. Introduction
The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...
- 1,776
1
vote
0
answers
85
views
"Barrier functions" in function spaces [closed]
In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
- 2,246
5
votes
0
answers
342
views
$N$-$th$ closed chain of six circles
Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
- 1,776
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
- 205
2
votes
0
answers
166
views
Pascal theorem for three dimensions
A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
- 1,776
2
votes
1
answer
375
views
Yiu's equilateral triangle-triplet points
In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...
- 1,776
6
votes
1
answer
273
views
Example of convex n-gon that cannot be decomposed into k congruent convex polygons
I asked a related question here on MO without any answers yet.
The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons.
...
- 15.8k
2
votes
1
answer
502
views
Feasibility Mixed integer Linear programming with quadratic constraints?
Consider the mixed integer program
$$Ax\leq b$$
$$By\leq c$$
$$\begin{bmatrix}x&y\end{bmatrix}C\begin{bmatrix}x\\y\end{bmatrix}+D\begin{bmatrix}x\\y\end{bmatrix}\leq d$$ where $x$ are integer ...
- 13.9k
-1
votes
2
answers
114
views
On OR condition in Linear Programming with exponentially many constraints [closed]
Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
- 13.9k
3
votes
1
answer
266
views
Strong polynomial algorithm for linear programming
What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
- 781
0
votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
- 13.2k
4
votes
1
answer
283
views
Reordering vertices of a polygon
Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...
user101163
10
votes
1
answer
329
views
Is there a triangle which makes dense set of angles by drawing medians?
This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
- 125
0
votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
1
vote
1
answer
170
views
Optimization of a continuous function
This is more like an optimization problem but any solution is appreciated.
I have a data set with input specifying power(demand) to be generated for a particular time period(TP).
Input:
Time --- ...
1
vote
0
answers
66
views
On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
- 13.9k
3
votes
1
answer
176
views
Relative to Isoperimetric inequality with n-polygon
Since Brahmagupta's formula and Bretschneider's formula we have the inequality:
Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic ...
- 1,776
2
votes
0
answers
75
views
Possible ordering of coordinates in a linear subspace [closed]
This question was asked on Mathematics Stack Exchange with no answers.
Let $X$ be a linear subspace of $\mathbb{R}^n$. For how many permutations $p$ on $1,...,n$ does there exists $x$ in $X$ with $...
- 31
9
votes
2
answers
595
views
Strengthened version of Isoperimetric inequality with n-polygon
Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
- 1,776
2
votes
1
answer
91
views
Linear program with one quadratic condition convex in domain of interest polynomial time solvable?
$c\leq xy$ is not a convex condition.
However we know $c\leq xy$ is convex in domain $x,y>0$ in $\mathbb R^2$ for any fixed $c\in\mathbb R$.
Is $c\leq x_1y_1+\dots+x_ny_n$ with $0\leq x_1,\dots,...
- 13.9k