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Questions tagged [convex-hulls]

For questions on the convex hull of a set, a set $X$ of points in a Euclidean space which is the smallest convex set that contains $X$. Consider adding (convex-analysis), or, for questions related to algorithms, (computational-geometry) and/or (discrete-geometry).

The convex hull of a set $X$ of points in a Euclidean space is the smallest convex set that contains $X$. They can be visualized by contracting a $n$-dimensional elastic sheet onto $X$.

Over $\mathbb{R}$ it is simply the set $[\min(X),\max(X)]$, over $\mathbb{R^2}$ it is a convex $n$-gon.

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If I stretch a convex polygon, does the original fit into the streched version?

Suppose you have a convex polygon $P=\mathrm{conv}(\{(x_1,y_1),\dots, (x_k,y_k)\})$ and you stretch it in one dimension, that is, we choose $\alpha>1$ and get a new polygon $P^\alpha=\mathrm{conv}(\{(\alpha x_1,y_1),\dots, (\alpha x_k,y_k)\})$. Is…
Dart
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Is the convex hull of closed set in $\mathbb R^{n}$ is closed?

Is convex hull of closed set in $\mathbb R^{n}$ closed?
user44174
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What shape is a Calippo?

The Calippo™ popsicle has a specific shape, that I would describe as a circle of radius $r$ and a line segment $l$, typically of length $2r$, that's at a distance $h$ from the circle, parallel to the plane the circle is on, with its midpoint on a…
SQB
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Non-trivial faces of the closed convex hull of a non-convex closed set with connected complement

I'm trying to prove or disprove a problem, but I'm struggling to make headway. Any help is appreciated. Suppose $X$ is a Hilbert Space, and $C \subseteq X$ is closed, bounded, non-convex, and $X \setminus C$ is connected. Does the boundary of…
Theo Bendit
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New Definition of Convex Hull

I have always defined the convex hull of a set $X$ to be the smallest convex set that contains $X$. I am currently reading about holomorphic convexity, and the author has introduced a new definition of the convex hull. That is, the convex hull of a…
user419593
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Convex hull of orthogonal matrices

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm not greater than one? It is easy to show that a convex combination of orthogonal matrices has norm (I mean the norm as…
Emanuele Paolini
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Meaning cone, ray, fan for polytopes

I'm trying to understand some mathematical operations and definitions for my project. Could you explain the meanings? $P_b =\{x∈\mathbb{R}^d : Ax≤b\}$ is a polytope. Let's have a 10x3 matrix $Ax≤b$ $$ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 &1 \\ 1 & 0…
Soner from The Ottoman Empire
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Convexifying a Concave Polygon by Reflections

Let $P$ be a given concave polygon and consider the following procedure. Take the convex hull $Q$ of $P$ and consider a side $AB$ of $Q$ which is not a side of $P$. Then, let $P'$ be the polygon obtained by $P$ by replacing the polygonal path of $P$…
Maurizio Barbato
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The image of every strictly non-expanding map between spheres is contained in an open hemisphere?

$\renewcommand{\S}{\mathbb{S}^n}$ $\renewcommand{\R}{\mathbb{R}}$ $\newcommand{\conv}{\operatorname{Conv}}$ $\newcommand{\Int}{\operatorname{Int}}$ This is a follow-up of this question. Let $f:A \subseteq \mathbb{S}^n \to \mathbb{S}^n$ be a map…
Asaf Shachar
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Probability that random variable falls inside convex hull of $n$ previous realizations

Let $X_1, ..., X_{n+1} \in \mathbb{R}^d$, $n > d$, be iid random variables following some distribution $F$ on $\mathbb{R}^d$. What is (a lower bound on) the probability that $X_{n+1}$ falls inside the (random) convex hull of $X_1, ..., X_n$? I found…
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Convex hull of rank-$1$ matrices is the nuclear norm unit ball

Let $$A := \left\{ u v^T : u \in \mathbb{R}^m, v \in \mathbb{R}^n, \|u\|_2 = \|v\|_2 = 1 \right\}$$ I would like to show that $$\textrm{conv}(A) = B_* := \left\{ X \in \mathbb{R}^{m \times n}: \|X\|_* = \textrm{Tr} \left( \sqrt{XX^T} \right) \le 1…
guanton
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Are convex objects determined by their silhouettes?

Informally, the silhouette of a 3D shape is a viewpoint-dependent 2D projection of it. You might imagine looking at several silhouettes and attempting to construct the overall shape. My question is theoretical: if you have access to the set of all…
user326210
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Geometric Intuition for Caratheodory's Theorem (for Convex Sets)

Consider the Wikipedia proof for Caratheodory's Theorem, the statement of which I have reproduced below. In short, I am looking for some geometric intuition about the modified coefficients in the proof, something that I may have been able to "see"…
Benjamin Bray
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Proof of Caratheodory's Theorem (for Convex Sets) using Radon's Lemma

I am self-studying some discrete geometry / convex analysis. Many descriptions of Caratheodory's Theorem for convex sets mention that Radon's Lemma can be used to simplify the proof, but I haven't seen it done. For reference, here is Radon's…
Benjamin Bray
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Test if point is in convex hull of $n$ points

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$, and I would like to check that some other point $y$ lies in their convex hull. How can I do this in some efficient way? I think that there was an algorithm based on checking the signs of pairwise inner…
Ulysses
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