Convex Hull

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tags: - colorclass/functional analysis ---The convex hull of a set of points in a Euclidean space ${\mathbb{R}}^d$ is a fundamental concept in computational geometry, optimization, and many areas of mathematics. It is the smallest convex set that contains all the points in the given set.

Definition

Given a set of points $X\subseteq {\mathbb{R}}^d$, the convex hull of $X$, denoted as $\text{conv}(X)$, is defined as the set of all convex combinations of points from $X$. Mathematically, it can be expressed as:

$$\text{conv}(X)=\left\{\sum _{i=1}^k{\lambda }_i{x}_i\mid x_i\in X,{\lambda }_i\ge 0,\sum _{i=1}^k{\lambda }_i=1,k\in \mathbb{N}\right\}$$

Here, each ${\lambda }_i$ represents a weight assigned to the point $x_i$, where all weights are non-negative and sum to 1.

Properties

1. Convexity: By definition, the convex hull is a convex set. 2. Compactness: If $X$ is a compact set, then $\text{conv}(X)$ is also compact. 3. Inclusion: For any set $X\subseteq {\mathbb{R}}^d$, $\text{conv}(X)$ is the smallest convex set that contains $X$.

Computation

The computation of the convex hull is a central problem in computational geometry. Several algorithms exist for this purpose, with their efficiency depending on the dimension $d$ and the number of points $n$. Common algorithms include:

- Graham’s scan and Andrew’s monotone chain algorithm: These are used for computing the convex hull in ${\mathbb{R}}^2$ and are based on sorting the points and then constructing the hull. - Quickhull: A method similar in concept to QuickSort and used in various dimensions, known for its average-case efficiency. - Incremental algorithms: These build the hull by adding one point at a time and are useful in higher dimensions.

Visual Example

In ${\mathbb{R}}^2$, if you consider a set of points, the convex hull can be visualized as the shape of a rubber band stretched around the outermost points. This shape includes all the given points and is convex.

Applications

The convex hull has applications in numerous fields: - Pattern Recognition: Defining the boundaries of object clusters. - Computer Graphics: Collision detection and shape analysis. - Robotics: Workspace determination and obstacle avoidance. - Geographic Information Systems (GIS): Defining boundaries around geographical features.

Further Exploration

The study of convex hulls is connected to several other mathematical topics and algorithms, such as Carathéodory’s theorem and Delaunay Triangulation. Understanding convex hulls can provide insights into the structure and behavior of complex systems and forms a foundation for advanced studies in discrete and computational geometry.