Questions tagged [convex-hulls]

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

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1answer
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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 ...
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Concentration of number of vertices in convex hull of random points

Dwyer showed that given a set of $n$ points $P$ which were uniformly and independently sampled from the unit cube $[0,1]^d$, if $X$ is the number of vertices in the convex hull of $P$, then $$\mathbb{...
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1answer
135 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,...
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1answer
59 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 ...
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1answer
215 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 ...
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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\...
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1answer
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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 ...
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58 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, ...
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1answer
109 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 $...
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1answer
361 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: ...
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1answer
134 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 (...
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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 ...
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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 ...
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2answers
574 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, ...
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1answer
163 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 ...
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1answer
462 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,...
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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 $...
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79 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 ...
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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 ...
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54 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 $...
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2answers
237 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, ...
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2answers
428 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( ...
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2answers
145 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, ...
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1answer
224 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 ...
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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 ...
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161 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/...
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1answer
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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 ...
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0answers
103 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 ...
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1answer
177 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)\...
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2answers
638 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 ...
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4answers
798 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 ...
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3answers
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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 ...
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3answers
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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}...
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3answers
863 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\...
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2answers
653 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?
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2answers
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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? ...
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3answers
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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 ...
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7answers
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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 ...