All Questions
1,547 questions
2
votes
1
answer
191
views
Optimization problem whose cardinality never exceeds 7 for some reason
I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
4
votes
1
answer
438
views
Characterization of $l_p$ up to a linear isometry
There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach spaces)...
- 7,426
10
votes
2
answers
676
views
Status of Barany's conjecture?
One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks:
Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$?
A convex $...
- 1,312
3
votes
0
answers
141
views
Covering fat objects with fat objects
The family of rectangles has the cover property, i.e.:
For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...
- 3,549
0
votes
1
answer
452
views
Large scale least squares of non symmetric and non square problems
Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...
- 125
13
votes
0
answers
406
views
Surface area of convex hull [duplicate]
Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
- 131
1
vote
1
answer
70
views
Heuristic for choosing n-vectors from n-sets
my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...
- 11
4
votes
0
answers
350
views
Convex functions with non-singular hessian measure are continuously differentiable?
It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
- 1,371
3
votes
1
answer
553
views
Calculate the discrete set of points B which are in the convex hull of the set of points A
This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
- 151
3
votes
3
answers
349
views
Sensitivity analysis in conic optimization
I have a conic optimization of the form:
$$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$
where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a ...
- 143
3
votes
0
answers
182
views
A question on Hilbert geometries as metric-measure spaces
Recall that a Hilbert geometry is the interior of a convex body $K \subset \mathbb{R}^n$ provided with the metric
$$
d(x,y) = \frac{1}{2} \ln\left(\frac{|x-b|}{|y-b|}\frac{|y-a|}{|x-a|}\right) ,
$$
...
- 13.5k
2
votes
0
answers
91
views
Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
1
vote
1
answer
267
views
Vertices of Curves and Eigenvectors of Hessian
This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to ...
- 115
1
vote
1
answer
152
views
extreme points of the image of a nonlinear vector-valued function
Consider a continuous function $f : D \rightarrow \mathbb{R}^m$, where $D \subseteq \mathbb{R}^n$ is a compact convex set. I am in search of a result that helps me say something about the extreme ...
- 183
6
votes
2
answers
408
views
Triangulations of special polyhedra
Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...
- 213
1
vote
2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 11
6
votes
2
answers
413
views
Random walk in a convex body or convex polytope
Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in
$\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside $\...
- 893
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
3
votes
1
answer
288
views
Generalization of notion of convexity
I am searching for the correct term for the following, if it exists.
A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius $...
22
votes
5
answers
1k
views
Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?
Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ...
- 1,303
10
votes
0
answers
787
views
Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?
I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...
3
votes
2
answers
306
views
Perimeter of a 'trapped' convex set
Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit below)....
- 115
2
votes
0
answers
111
views
Quickly checking an inequality on a convex region
I previously posted this question to math.sx at: https://math.stackexchange.com/questions/748015/quickly-checking-if-an-inequality-holds-on-a-convex-region but I'm thinking that MO may be more ...
- 21
4
votes
1
answer
750
views
submatrix of a given size with maximum frobenius norm
Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
- 859
5
votes
2
answers
165
views
Computational complexity of deciding isomorphism of rational polyhedral cones
Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...
- 38.6k
3
votes
0
answers
107
views
Decomposing a cone based on decompositions of its facets
Let $C$ be a cone in $\mathbb{R}^d$, and let $x_1, \dots, x_k$ be its extreme rays. Suppose that the $x_i$ satisfy:
For all $i, j$, $\langle x_i, x_j \rangle \ge 0$,
There is a partition $A \cup B = ...
- 413
2
votes
0
answers
120
views
integrality of a linear program -- binary equality constaints
Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...
- 127
4
votes
1
answer
3k
views
optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
- 49
1
vote
0
answers
91
views
Tubular neighbourhood which is nowhere piecewise linear
I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
- 11
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
9
votes
0
answers
380
views
An affine invariant of convex bodies
The following invariants of "pointed" convex bodies (i.e., pairs consisting of a convex body and a distinguished point in its interior) roughly measure how many of its linear images fit between the ...
- 13.5k
3
votes
0
answers
157
views
Intrinsic volumes of a family of convex sets $\{K_n\}$
Consider a family of convex sets $\{K_n\}$ such that $K_n \subset \mathbb{R}^n$ for each $n$. The kinds of sets one might be considering could be, for instance,
$K_n$ is the cube of side $2A$, i.e., ...
- 1,034
6
votes
1
answer
530
views
References to proofs of a theorem by Van Kampen-Flores
Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other.
This ...
6
votes
0
answers
160
views
Norms and distributions
Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := \...
- 13.5k
2
votes
1
answer
269
views
Homotopy with non piece-wise linear boundary
in the middle of a long proof I encounter the following problem.
Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand $E$...
- 21
4
votes
2
answers
212
views
combinatorial and linear duality
Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\...
- 987
2
votes
1
answer
134
views
Integer point in a non-empty polytope
I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...
- 243
5
votes
4
answers
905
views
Probability that convex hull of multivariate Gaussian sample contains a given point
I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls ...
- 101
1
vote
0
answers
75
views
Are there any known bounds on the value of solutions of linear integer programming?
Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...
- 111
11
votes
2
answers
1k
views
Triangle with largest perimeter in a convex region
What is the largest value of $r$ such that the following statement is always true?
"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least ...
- 111
4
votes
2
answers
214
views
The maximal discrete parallelepiped in a convex body
Does the positive constant $c_d$, depending only from dimension, with the following property exist?
Property:
for every convex body $K\subset \mathbb R^d$ there exists parallelepiped $P\subset K$ ...
1
vote
0
answers
413
views
Extreme points of a set related to semidefinite cone
Let $X \in \mathbb{R}^{n \times n}$ be symmetric matrix. Consider the following set
$$
\mathcal{C} = \{ X: X \succeq 0, \quad 0 \le X_{ij} \le 1, \forall i,j\}
$$
What are the extreme points of this ...
- 1,731
13
votes
0
answers
495
views
Unit ball of smallest volume in a Hilbert geometry
In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body ...
- 13.5k
1
vote
2
answers
282
views
Extreme points and centroid
It is well-known that the centroid of a triangle is the intersection point of its three medians. The medians happen to be area bisectors, but it seems that most (all?) other lines through the ...
- 29.2k
2
votes
0
answers
101
views
Measure of points with small neighborhood in convex bodies
Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e ...
1
vote
1
answer
4k
views
Maximizing linear objective function with absolute values
This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function ...
- 25.4k
2
votes
2
answers
841
views
Finding the maximum of a multivariate polynomial of degree one
I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...
user24451
3
votes
1
answer
804
views
Approximation of a convex body by a contained polytope
This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...
- 591
1
vote
0
answers
493
views
Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
14
votes
0
answers
310
views
How large are the smallest-area projections of a high-dimensional convex body?
Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that
$$\intop_B x \, dx = 0$$
$$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$
a ...
- 4,000