All Questions
15 questions
5
votes
2
answers
294
views
Convex caps with prescribed edges
Let $P$ be a convex polygon in the plane $R^2=R^2\times \{0\}$, and $E$ be the edge graph of some subdivision of $P$ into convex polygons, which is $3$-connected. Does there exist a convex polyhedral ...
- 7,194
5
votes
2
answers
153
views
Expressing a convex Polytope as a sublevel set of a function
Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to \...
- 1,893
3
votes
0
answers
51
views
testing whether a polyhedral complex is convex
Definitions
A (polyhedral) cone in $\Bbb R^n$ is the solution set of a finite number of inequalities of the form $a_1x_1+\cdots+a_nx_n\geq 0$. Note that I don't require strict convexity, i.e. a cone $...
- 3,079
2
votes
1
answer
113
views
Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points
Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta ...
- 6,853
2
votes
1
answer
349
views
Several convex polytopes in a simplex; fix an extreme point for each; how many can be supported by a function monotonic on all line segments?
Sorry the title may be unclear. I do not know how to give it a good title.....
Let $\Delta$ be a probability simplex of $R^N$; i.e. set of all points $x$ such that $x\geq0$ and $\sum_{k=1}^Nx^k\leq1$....
- 159
1
vote
2
answers
431
views
Higher dimensional convex hull
Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as $E(v)=...
user33954
1
vote
1
answer
391
views
Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces
Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...
- 2,689
1
vote
1
answer
1k
views
Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball [closed]
Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the ...
- 11
1
vote
0
answers
43
views
Vertex enumeration for polytope with a sparse halfplane description?
Say I have a (bounded convex) polytope $P\subset\mathbb R^d$ with description $Ax\le b$, where $A$ is sparse in the sense that there are at most $k$ nonzero entries in each row or column, where $k$ is ...
- 523
1
vote
0
answers
52
views
How does one translate from convex hull to a set of facets (inequalities)? [duplicate]
Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...
- 11
1
vote
0
answers
48
views
Efficient scissors congruence between efficiently describable convex polytopes and simplex?
Is there a convex polytope in $\mathbb R^n$ describable by only $O(poly(\log n))$ half-plane inequalities with positive volume (so at least $n+1$ vertices) such that the standard simplex has a ...
- 1,826
1
vote
0
answers
65
views
Covering a simplex efficiently by efficiently describable polytopes?
Take a standard simplex or cube in $\mathbb R^n$.
Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities?
If not what ...
- 1,826
1
vote
0
answers
64
views
Polytopes that can be efficiently described and efficiently covered by cubes or simplices?
Is there a bounded convex polytope $\mathcal P\subseteq\mathbb R^n$ with $m$ vertices, whose vertex vectors span $\mathbb R^n$ (so $m$ is $\Omega(n)$) and just $O(poly(\log n))$ half-plane ...
- 1,826
0
votes
0
answers
37
views
Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
0
votes
0
answers
52
views
Efficient sampling from a polytope with large number of contraints [duplicate]
As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...
- 6,853