All Questions
21 questions
0
votes
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69
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Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces
Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
- 1,467
2
votes
1
answer
159
views
Conic hull of a rectangle
I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
- 275
1
vote
0
answers
61
views
Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
0
votes
1
answer
101
views
Estimation via projecting onto a convex body
Assume that $\theta$ is in a convex body $K \in \mathbb{R}^n$ and we observe $y = \theta + z$, where $z$ is a noise term (following, say, the normal distribution). Consider an estimator of $\theta$ by ...
- 773
0
votes
0
answers
118
views
Weak derivative of projection onto probabilist's simplex
Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
- 5,405
4
votes
1
answer
194
views
How to solve this minimax matrix optimization problem?
Recently, I want to know how well can a $\ell_1$ ball be approximated by the image of a $\ell_2$ ball under a linear transform. I formulate this problem as the following optimization problem.
\begin{...
- 550
1
vote
1
answer
337
views
High-dimensional polytopes
I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex:
How can ...
- 43
4
votes
0
answers
229
views
How to find the dimension of the polar cone of a convex cone generated by some given vectors
Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...
- 461
1
vote
0
answers
99
views
Finding a point on a convex set
Given a compact bounded convex set $\mathcal C\subseteq\mathbb R^n$ given by $t$ hyperplane inequalities I want to find a point $u\in\mathcal C$ such that for all $v\in\mathcal C$
a convex relation $...
- 1,826
0
votes
1
answer
117
views
Closed form solutions for maximal subsets of convex polytopes
I'm looking for any known exact results about inscribing simple convex bodies inside a convex polytope. The most famous is the Löwner-John ellipsoid, but as far as I understood in general there is no ...
- 445
3
votes
1
answer
258
views
Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
- 422
1
vote
1
answer
266
views
Relative interior of a normal cone at a face of a convex polytope?
Suppose $A$ is a nonempty convex polytope in $\mathbb{R}^n$. Suppose $F$ is a face of $A$.
Consider the normal cone of $A$ at $F$:
$C_A(F)=\{v\in\mathbb{R}^n:v\cdot x\geq v\cdot y\ \forall\ x\in ...
- 159
1
vote
0
answers
53
views
Projecting two convex polyhedra onto their intersection
Suppose we are given two convex polyhedra $\mathcal{C}_1, \mathcal{C}_2 \subset \mathbb{R}^n$ with non-empty intersection $\mathcal{C}_1 \cap \mathcal{C}_2 \neq \emptyset$.
For the orthogonal ...
- 337
2
votes
0
answers
62
views
on vectors for which the intersection of their convex hull and the nonegative orthant is the unit simplex
Consider the vectors $r^1 = (0,2,-1)$, $r^2 = (-1,0,2)$, and $r^3 = (2,-1,0)$. Two properties of these vectors that interest us here are:
1) The $i$'th coordinate of $r^i$ is 0, and
2) The ...
- 745
3
votes
1
answer
452
views
How to find extreme points of a set related to Minkowski's Theorem?
Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^n$. For $m>n$, we can define $\Lambda$ to be the set
$$\{(\lambda_1, ..., \lambda_m):\sum_{i=1}^m \lambda_i=1, \lambda_i\ge0, and \mbox{ there exist}...
- 1,350
1
vote
0
answers
168
views
Projecting on a a special polyhedron
Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$
Notice that $\mathcal P_X$ is symmetric about the origin.
...
- 6,853
1
vote
0
answers
77
views
Projecting on a convex compact polytope with special form
Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
- 6,853
2
votes
0
answers
210
views
Finding optimal linear transformation for intersection of convex polytopes
I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) ...
- 695
4
votes
1
answer
3k
views
Find the minimum distance between two convex hulls
We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...
- 233
0
votes
1
answer
124
views
polynomial expression for counting number of integral points of a set
Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...
- 1,713
4
votes
2
answers
981
views
Convex Sets and Nearest Neighbors
For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is unique. It is known that $c(x) = s$ ...
- 177