🔎Convex Geometry Unit 2 – Convex Hulls and Extreme Points

Key Concepts and Definitions

• Convex set a set of points where any two points can be connected by a line segment that lies entirely within the set
• Convex hull the smallest convex set that contains all points in a given set, often visualized as a rubber band stretched around the outermost points
• Extreme point a point in a convex set that cannot be expressed as a convex combination of other points in the set
• Supporting hyperplane a hyperplane that intersects a convex set and has the entire set on one side or the other
  • Exists at every boundary point of a convex set
• Convex combination a weighted average of points where the weights are non-negative and sum to 1
• Affine hull the smallest affine subspace that contains a given set of points
• Convex polytope a convex hull of a finite set of points in Euclidean space

Geometric Properties of Convex Hulls

• Convexity preserved under intersection the intersection of two convex sets is always convex
• Convex hull unique for a given set of points, there exists a unique convex hull
• Convex hull contains all points in the set the convex hull is the smallest convex set that includes all points in the original set
• Extreme points form the boundary of the convex hull the convex hull is the convex combination of its extreme points
• Convex hull may have fewer dimensions than the ambient space (a line segment in 3D space)
• Convex hull is a closed set it contains all its boundary points
• Convex hull is the intersection of all half-spaces containing the set half-spaces are defined by supporting hyperplanes

Algorithms for Constructing Convex Hulls

• Gift wrapping algorithm (Jarvis march) starts with an extreme point and "wraps" around the set, finding the next extreme point at each step
  • Time complexity: $O(nh)$ for $n$ points and $h$ extreme points
• Graham scan sorts points by angle from a chosen point, then performs a stack-based scan to construct the hull
  • Time complexity: $O(n \log n)$ due to sorting
• Quickhull recursively divides the point set, discarding interior points and combining the hulls of the subsets
  • Average time complexity: $O(n \log n)$, worst case: $O(n^2)$
• Divide-and-conquer hull merges the hulls of two separated point sets, eliminating interior points
  • Time complexity: $O(n \log n)$
• Incremental hull adds points one at a time, updating the hull with each addition
• Chan's algorithm combines the gift wrapping and Graham scan algorithms for improved efficiency
  • Time complexity: $O(n \log h)$

Extreme Points: Characteristics and Significance

• Extreme points cannot be expressed as convex combinations of other points in the set
• Extreme points are the vertices of the convex hull polytope
• Extreme points lie on the boundary of the convex hull
• Supporting hyperplane exists at each extreme point, separating it from the rest of the set
• Krein-Milman theorem a compact convex set is the convex hull of its extreme points
• Extreme points play a crucial role in linear programming and optimization
  • Optimal solutions often found at extreme points
• Identifying extreme points helps simplify convex hull computation by discarding interior points

Applications in Optimization and Computer Graphics

• Linear programming optimal solutions often found at extreme points of the feasible region (convex polytope)
• Convex optimization techniques (gradient descent) can efficiently find solutions within convex sets
• Collision detection in computer graphics convex hulls used for simplified object representation and efficient intersection tests
• Path planning in robotics convex hulls help define obstacle-free regions and navigate through environments
• Shape analysis and pattern recognition convex hulls capture essential shape features and enable shape comparison
• Computational geometry convex hulls fundamental in many geometric algorithms and data structures (Voronoi diagrams, Delaunay triangulations)
• Machine learning convex hulls used in support vector machines (SVMs) for classification and regression tasks

Theoretical Foundations and Proofs

• Caratheodory's theorem any point in a convex hull can be represented as a convex combination of at most $d+1$ points, where $d$ is the dimension of the space
• Radon's theorem any set of $d+2$ points in $d$-dimensional space can be partitioned into two subsets whose convex hulls intersect
• Helly's theorem if a collection of convex sets in $d$-dimensional space has the property that any $d+1$ of them have a common point, then all of the sets have a common point
• Separating hyperplane theorem for any two disjoint convex sets, there exists a hyperplane that separates them
• Minkowski's theorem a convex set is the sum of its extreme points and its recession cone
• Farkas' lemma provides a connection between the solvability of a linear system of inequalities and the existence of a solution to a related system of linear equations

Practical Examples and Problem-Solving

• Finding the smallest bounding box for a set of points compute the convex hull and find the minimum and maximum coordinates in each dimension
• Identifying outliers in data points that lie far outside the convex hull of the majority of the data may be considered outliers
• Solving linear programming problems graphically the optimal solution is found at an extreme point of the feasible region (defined by constraint inequalities)
• Constructing Voronoi diagrams the convex hull of the input points is often used as a starting point for the algorithm
• Collision avoidance in robotics represent obstacles as convex hulls and plan paths that avoid these regions
• Designing efficient packaging or container storage arrange items to minimize the volume of their convex hull
• Image processing convex hulls used for shape approximation, feature extraction, and morphological operations

Advanced Topics and Extensions

• Approximate convex hulls finding a convex set that closely approximates the input set, often used for large or high-dimensional datasets
• Convex hull algorithms in higher dimensions the complexity of convex hull algorithms grows exponentially with the dimension of the space
• Dynamic convex hulls maintaining the convex hull of a set of points that can be added or removed over time
• Weighted convex hulls assigning weights to points and computing the convex hull with respect to these weights
• Convex hull of moving points tracking the convex hull of a set of points that change position over time
• Convex hull in non-Euclidean spaces (hyperbolic, spherical) adapting convex hull concepts and algorithms to different geometric spaces
• Duality between convex hulls and halfspace intersections a point in one set corresponds to a halfspace in the dual set, and vice versa