# Questions tagged [convex-hulls]

The convex hull of a set $X$ of points in a Euclidean space is the smallest convex set that contains $X$.

43
questions

3
votes

0
answers

46
views

### Volume of all Voronoi cells in n-dimensional bounded space

How can one find the volume of all Voronoi cells (bounded and unbounded) in an $n$-dimensional bounded space? For instance, consider an $N$-dimensional space (hypercube) with bounds on each dimension ...

1
vote

1
answer

90
views

### If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?

That is, if $P$ is a finite set, and $C$ is the set of points in $P$ which lie on the boundary of the convex hull of $P$, then is $P$ contained in the convex hull of $C$?
It seems true intuitively. In ...

1
vote

0
answers

64
views

### On known links between convexity and fuzzy logic

The following are thoughts I had during a joint work with P. Olver on maps from spheres to the convex hull of finitely many points in some finite-dimensional vector space. I do not wish to discuss the ...

8
votes

1
answer

202
views

### Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...

14
votes

7
answers

2k
views

### Finite set of non-collinear points on plane with every point having ≥ 3 equidistant neighbors? [closed]

Does there exist a finite set of points on the Euclidean plane, such that:
No 3 points are collinear, and
Every one of the points has (at least) three other points in the set at the same distance ...

0
votes

0
answers

55
views

### Given m vectors in n dimension where m>>>n, how do you find the vectors that define the largest convex hull constructed with the vectors?

Say there are m vectors in n dimensional space (m>>>n).
There exists a largest convex hull defined by a subset of those vectors.
My goal is to describe the space that is strictly inside the ...

2
votes

1
answer

104
views

### Commutation of linear maps and extreme points

Let $X,Y$ be real vector spaces, $T: X\to Y$ be a linear map, and fix a nonempty $S\subseteq X$ (we do not assume that $S$ is neither convex nor compact (indeed, right now we do not assume any ...

6
votes

1
answer

216
views

### Convex hull of a variety in real space

I am a physicist currently working on a question posed as part of an algebraic geometric description of a physical set:
I did not find a question that is closely related to what I am searching for yet,...

0
votes

1
answer

117
views

### How hard is a linear programming with a bounded constraint?

Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...

1
vote

1
answer

853
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 ...

2
votes

2
answers

252
views

### Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference.
Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...

8
votes

1
answer

1k
views

### Two questions on the permutohedron

The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes.
I ...

2
votes

0
answers

66
views

### Convex hull of prefix sum of $n$ ordered random points

Suppose we have $n$ ordered realizations of a random variable uniformly distributed over the unit cube $P = (p_1, p_2, \cdots, p_n), p_i \in [0,1]^d $. And we obtain the prefix sum $S = (p_1, p_1+p_2, ...

2
votes

1
answer

160
views

### Polytope with most faces

Fix $m,n \in \mathbb{N}$ with $m \ge n+1$. Take $m$ points in general position in $\mathbb{R}^n$ and let $P$ be their convex hull. What is the maximal number of (external, codimension-one) faces that $...

5
votes

1
answer

476
views

### How to apply Hahn-Banach to the convex hull?

I am trying to understand the proof of Lemma 4.1.2 in Michel Talagrand's publication from 1995 on concentration inequalities (see below for the precise question statement):
A bit of context: ...

2
votes

1
answer

148
views

### Number of orthants intersected by a convex hull

I'm trying to figure out the following problem:
Let $x_1,\ldots,x_k\in\mathbb{R}^n$ be some points for some $k<n$. Let $\mbox{conv} (x_1,\ldots,x_k)$ be their convex hull. I'm looking for a tight (...

0
votes

0
answers

31
views

### Counterexample to a generalization of planar convex hulls

I am looking for a counter example to the following conjecture:
for every finite compact region $\mathcal{R}\subset\mathbb{R}^2$ that is defined by a rectifiable Jordan curve
and every fixed and ...

1
vote

0
answers

49
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 ...

4
votes

2
answers

770
views

### Convex hull in a discrete space [closed]

I know some algorithms which compute the convex hull in a continuous space.
Are there efficient algorithms to compute it in a discrete domain?
For example in 3D discrete space, given the blue points, ...

2
votes

1
answer

249
views

### Algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane

I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on ...

16
votes

1
answer

556
views

### Convex hull of all rank-$1$ $\{-1, 1\}$-matrices?

Consider the set $\mathbb{R}^{m \times n}$ of $m \times n$ matrices. I am particularly interested in properties of polytope $P$ defined as a convex hull of all $\{-1,1\}$ matrices of rank $1$, that is,...

1
vote

0
answers

41
views

### Algorithm for Calculating Spheric Convex Hulls of Finite Pointsets

Let the Spheric Convex Hull ($\mathrm{CH}_S$) denote the intersection of all closed spheres that contain a compact $\Sigma\subset\mathbb{R}^n$ and on their boundary at least $n+1$ distinct points of $...

1
vote

0
answers

83
views

### Compute the edge-skeleton of a polytope given by its vertices

Let $P$ be a polytope given by a vertex description, i.e., $P=conv(\{x_1,\ldots,x_m\})\subset\mathbb{R}^n$.
Is there an efficient (i.e., not relying on Linear Programming) algorithm to compute the ...

2
votes

0
answers

114
views

### Convex hull of piece-wise linear functions

Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact ...

1
vote

0
answers

59
views

### Are the sets whose convex hull surface admits multiple representations a shy set of sets?

Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...

2
votes

2
answers

285
views

### How to show that the origin is not in the convex hull, in this problem?

I am given $3$ points $(x_i,y_i,z_i) \in \mathbb{R}^3\setminus \{\mathbf{0} \}$, for $i=1,2,3$, satisfying the following two polynomial equations (the first equation is actually not the intended one, ...

11
votes

2
answers

517
views

### Convex hull of the Stiefel manifold with non-negativity constraints

Consider the Stiefel manifold
$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$
where $I_k$ is the $k$-dimensional identity matrix. It is well known that
$$\mathrm{conv} \left( ...

6
votes

2
answers

148
views

### Inequality on permutation polytope

Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n), c = (c_1, \cdots, c_n) \in \mathbb{R}^n$ with $a_1 \geq \cdots \geq a_n, b_1 \geq \cdots \geq b_n, 0 < c_1 \leq \cdots \leq c_n$.
In addition, ...

0
votes

1
answer

302
views

### Finding the minimum-area parallelogram containing all vertices of a cuboid projected to a plane

Description edited after comments:
Let's say I have a 3d cuboid that I projected onto a 2d plane. Now I want to find the minimum-area parallelogram that contains all those projected vertices. How do ...

1
vote

0
answers

92
views

### quick hull algorithm detail

When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...

1
vote

0
answers

183
views

### Facet Enumeration Problem nondegeneracy case

Hello in case of a nondegeneracy case of the Facet Enumeration Problem, there is a polynomial algorithm for the convex hull problem as written here https://www.inf.ethz.ch/personal/fukudak/polyfaq/...

6
votes

1
answer

2k
views

### Approximation of convex hull in high dimension

What are efficient methods (polytime) to compute an approximation of the convex hull in high dimension (say, $30000$) for a given set of points?
Edit:
I am looking for an algorithm for getting the ...

6
votes

0
answers

116
views

### Convex hull of all-ones principal submatrices

For a subset $S$ of $\{1,\ldots,n\}$,
let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of ...

3
votes

1
answer

201
views

### How to show it is contained in a convex hull?

There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F_i)\...

25
votes

2
answers

677
views

### Convex hull of total orders

Let $n$ be a positive integer and $\prec$ an arbitray total order on $\{1,\dots,n\}$. I associate to this order a vector $v$ with one coordinate for every pair $(i,j)$ s.t. $1\leq i\neq j \leq n$, by ...

10
votes

4
answers

979
views

### Convex Hull of Path Connected sets

This is a pretty easy question to ask, but haven't seen it anywhere.
Suppose I have some continuous path $X$ in $\mathbb{R}^n$ and I want to get the convex hull of $X$, $\operatorname{co}(X)$.
Is it ...

16
votes

3
answers

4k
views

### Convex hull on a Riemannian manifold

Let $M$ be a complete Riemannian 2-manifold.
Define a subset $C$ of $M$ to be convex
if all shortest paths between any two points
$x,y \in C$ are completely contained within $C$.
For a finite set of ...

4
votes

3
answers

2k
views

### Minimum norm of convex hull

I am currently stuck at a problem which seems too easy to be stuck at to me...
Summary
Let $H$ be the convex hull of the points $d_1,\ldots, d_n\in \mathbb{R}^d$. How can one compute
\[\min_{x\in H}...

4
votes

3
answers

1k
views

### Holomorphically Convex Hull a Subset of the convex hull of

This comes from Hörmander's "An Introduction to Complex Analysis in Several Variables".
We defined the $A(\Omega)$-hull (analytic functions in an open set $\Omega$). $\hat{K}$ of a compact set $K\...

2
votes

2
answers

679
views

### Where to submit a new convex hull algorithm?

Recently, I devised a new convex hull algorithm. Is there any forum where I can submit my work?

12
votes

2
answers

2k
views

### Convex hull of $k$ random points

Suppose we have $k$ realizations of a random variable uniformly distributed over the unit cube $[0,1]^n$.
What is the probability that their convex hull has all of the $k$ points as extreme points?
...

18
votes

3
answers

3k
views

### Deciding membership in a convex hull

Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...

27
votes

8
answers

5k
views

### Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset.
Is it true that convex hull of $K$ is compact?
Comments:
Convex hull of $K$ = intersection of all closed convex sets ...