Glossary

2.1 Convexity and convex sets

6 min read•august 21, 2024

Convexity is a fundamental concept in computational geometry, simplifying complex problems and enabling efficient algorithms. It's all about shapes without dents or caves, like circles and triangles, which have special properties that make calculations easier.

Convex sets contain all line segments between any two points inside them. This simple idea leads to powerful mathematical tools and algorithms for solving geometric problems, from finding the smallest shape enclosing a set of points to detecting collisions in video games.

Definition of convexity

  • Convexity plays a crucial role in computational geometry by simplifying complex geometric problems and enabling efficient algorithms
  • Convex shapes and sets exhibit unique properties that allow for optimized computational approaches in various geometric calculations and analyses

Convex sets vs non-convex sets

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  • Convex sets contain all line segments connecting any two points within the set
  • Non-convex sets have at least one line segment between two points that lies partially outside the set
  • Convex sets form a single, unbroken region without any indentations or holes
  • Visualize convex sets as shapes without any "dents" or "caves" (circles, triangles, rectangles)

Mathematical formulation of convexity

  • Defined using the concept of line segments between any two points in the set
  • Formally expressed as ∀x,y∈S,∀t∈[0,1]:tx+(1−t)y∈S\forall x,y \in S, \forall t \in [0,1] : tx + (1-t)y \in S
  • Utilizes the notion of to describe all points within the set
  • Applies to both finite-dimensional vector spaces and infinite-dimensional function spaces

Properties of convex sets

  • Convex sets exhibit fundamental characteristics that make them valuable in computational geometry
  • Understanding these properties enables efficient algorithms for various geometric problems and optimizations

Intersection of convex sets

  • Always results in another or an empty set
  • Preserves convexity, allowing for simplified intersection computations
  • Enables efficient algorithms for finding common regions between multiple convex shapes
  • Applies to both 2D and higher-dimensional spaces (planes intersecting in 3D space)

Union of convex sets

  • Generally does not preserve convexity
  • May result in non-convex shapes with complex geometries
  • Requires additional processing to maintain convexity ( computation)
  • Presents challenges in computational geometry when dealing with multiple objects

Convex combinations

  • Linear combinations of points in a convex set with non-negative coefficients summing to 1
  • Expressed mathematically as ∑i=1nαixi\sum_{i=1}^n \alpha_i x_i where ∑i=1nαi=1\sum_{i=1}^n \alpha_i = 1 and αi≥0\alpha_i \geq 0
  • Forms the basis for defining convexity and understanding convex set properties
  • Enables the representation of any point within a convex set as a combination of its vertices

Types of convex sets

  • Convex sets encompass various geometric shapes and structures in computational geometry
  • Understanding different types of convex sets aids in selecting appropriate algorithms and representations

Convex polygons

  • Closed 2D shapes with straight sides and no internal angles greater than 180 degrees
  • Characterized by vertices always pointing outwards
  • Efficiently represented by an ordered list of vertices
  • Include regular polygons (equilateral triangles, squares, regular hexagons)

Convex polyhedra

  • 3D objects with flat faces, straight edges, and vertices pointing outwards
  • Generalization of to three-dimensional space
  • Represented by sets of vertices, edges, and faces
  • Encompass familiar shapes (cubes, tetrahedra, octahedra)

Convex hulls

  • Smallest convex set containing a given set of points
  • Computed using algorithms (Graham scan, , QuickHull)
  • Serves as a fundamental operation in many computational geometry problems
  • Finds applications in shape analysis, collision detection, and clustering

Algorithms for convex sets

  • Computational geometry employs various algorithms to manipulate and analyze convex sets
  • These algorithms form the foundation for solving complex geometric problems efficiently

Convex hull algorithms

  • Graham scan algorithm operates in O(n log n) time complexity
    • Sorts points based on polar angle and builds the hull incrementally
  • Jarvis march (gift wrapping) algorithm has O(nh) time complexity
    • Iteratively selects the next point with the smallest right-hand turn
  • uses divide-and-conquer approach
    • Recursively partitions points and constructs the hull

Point-in-polygon tests

  • Ray casting algorithm counts intersections of a ray with polygon edges
  • Winding number algorithm computes the number of times a polygon winds around a point
  • Utilizes convexity properties for optimized testing in convex polygons
  • Applies to both convex and non-convex polygons with different efficiency levels

Intersection detection

  • (SAT) efficiently detects intersections between convex shapes
  • (BSP) trees organize space for faster intersection queries
  • (BVH) use nested bounding volumes to accelerate intersection tests
  • Exploits convexity properties to simplify and speed up collision detection algorithms

Applications of convexity

  • Convexity concepts find widespread use in various computational geometry applications
  • Understanding these applications highlights the practical importance of convex sets

Optimization problems

  • relies on convex feasible regions for efficient solutions
  • Convex guarantee global optima, simplifying solution methods
  • algorithms converge to global minima in convex functions
  • Applies to resource allocation, network flow, and machine learning problems

Collision detection

  • Separating Axis Theorem (SAT) efficiently detects collisions between convex objects
  • Bounding volume hierarchies use convex shapes (spheres, axis-aligned bounding boxes) for broad-phase collision detection
  • GJK (Gilbert-Johnson-Keerthi) algorithm computes distance between convex shapes
  • Enables real-time physics simulations in video games and robotics

Computer graphics

  • Convex decomposition simplifies complex 3D models for efficient rendering
  • Occlusion culling uses convex hulls to determine object visibility
  • Shadow volume algorithms utilize for real-time shadow rendering
  • Convex hull computation aids in level-of-detail generation for 3D models

Convexity in higher dimensions

  • Convexity concepts extend beyond three dimensions in computational geometry
  • Higher-dimensional convex sets present unique challenges and properties

Hyperplanes and halfspaces

  • Hyperplanes generalize the concept of planes to higher dimensions
  • Defined by linear equations in n-dimensional space
  • Halfspaces represent all points on one side of a hyperplane
  • Form building blocks for describing convex sets in higher dimensions

Convex polytopes

  • Higher-dimensional analogs of convex polygons and polyhedra
  • Bounded by hyperplanes in n-dimensional space
  • Vertex representation becomes inefficient as dimensionality increases
  • Facet representation often preferred for high-dimensional polytopes

Computational complexity

  • Analyzing the efficiency of convex set algorithms informs algorithm selection and problem-solving approaches
  • Complexity considerations become crucial as problem sizes and dimensions increase

Time complexity of convex algorithms

  • Convex hull algorithms vary in complexity (O(n log n) for Graham scan, O(n^2) for Jarvis march)
  • Linear programming algorithms (Simplex, Interior Point) have different worst-case and average-case complexities
  • Intersection detection algorithms often have O(n log n) complexity for preprocessing and O(log n) for queries
  • Dimensionality significantly impacts algorithm performance in higher-dimensional problems

Space complexity considerations

  • Convex hull storage requires O(n) space for output in 2D and 3D cases
  • Higher-dimensional convex sets may require exponential space to represent explicitly
  • Trade-offs between time and space complexity often arise in convex set algorithms
  • Implicit representations and approximation techniques help manage space complexity in high dimensions

Duality and convexity

  • Duality concepts in computational geometry often involve convex sets
  • Understanding duality provides alternative perspectives on geometric problems

Polar duality

  • Maps points to hyperplanes and vice versa in projective geometry
  • Preserves incidence relationships and convexity properties
  • Transforms convex polytopes into their polar duals
  • Enables solving certain geometric problems by working in the dual space

Voronoi diagrams

  • Partition space based on proximity to a set of points or objects
  • Dual to Delaunay triangulations, which maximize the minimum angle of triangles
  • Voronoi cells for convex sites are always convex
  • Find applications in nearest neighbor searches and computational biology

Convexity in machine learning

  • Convex optimization plays a crucial role in many machine learning algorithms
  • Convexity properties enable efficient training and guarantee global optima

Support vector machines

  • Utilize convex optimization to find the optimal separating hyperplane
  • Kernel trick extends SVM to non-linearly separable data while maintaining convexity
  • Soft-margin SVM formulation results in a convex quadratic programming problem
  • Convexity ensures efficient training and convergence to global optima

Convex optimization techniques

  • Gradient descent converges to global minima for convex functions
  • Stochastic gradient descent (SGD) efficiently handles large-scale machine learning problems
  • Interior point methods solve convex optimization problems with inequality constraints
  • Convex relaxation techniques approximate non-convex problems as convex ones for tractable solutions

Key Terms to Review (27)

Binary Space Partitioning: Binary space partitioning is a method for recursively dividing a space into two half-spaces by hyperplanes, which is particularly useful in computer graphics, computational geometry, and spatial data structures. This technique helps in organizing objects within a given space, allowing for efficient rendering, collision detection, and other operations by breaking down complex scenes into manageable parts. The partitions created can be either convex or non-convex, but the relationship to convexity arises when considering how objects within those partitions interact with each other and how their spatial relationships can be simplified.
Bounding volume hierarchies: Bounding volume hierarchies (BVHs) are tree structures that organize objects in a spatial environment, allowing for efficient collision detection and ray tracing in computer graphics and computational geometry. They work by enclosing objects within simple geometric shapes, or bounding volumes, which can be recursively subdivided into smaller volumes, enabling quick rejection of pairs of objects that do not intersect. This concept relates to the applications of convex hulls as they often utilize BVHs to optimize spatial queries and collision detection processes, while also being grounded in the principles of convexity and convex sets.
Carathéodory's Theorem: Carathéodory's Theorem states that if a point lies in the convex hull of a set of points in a Euclidean space, then it can be expressed as a convex combination of at most 'd+1' points from that set, where 'd' is the dimension of the space. This theorem provides a foundational understanding of how convex combinations work and connects deeply with concepts like convexity, the Minkowski sum, and approximating convex hulls, highlighting the structure of convex sets in higher dimensions.
Convex combinations: A convex combination is a linear combination of points where the coefficients are non-negative and sum up to one. This concept is essential in understanding convex sets, as it describes how points can be blended together to form new points that lie within the same set. It highlights the idea that if you take two or more points from a convex set and mix them appropriately, you will always end up with another point in that same set, reinforcing the definition of convexity.
Convex Hull: The convex hull of a set of points is the smallest convex polygon that can enclose all the points in that set. This concept is important as it helps to define the boundary of a shape formed by a collection of points, providing a foundational element in various computational geometry algorithms and applications.
Convex polygons: Convex polygons are simple polygons in which all interior angles are less than 180 degrees, meaning that any line segment drawn between two points inside the polygon remains inside it. This property ensures that the shape bulges outward, and no angles point inward, making convex polygons crucial in various geometric computations and algorithms.
Convex polyhedra: Convex polyhedra are three-dimensional geometric shapes formed by flat polygonal faces, straight edges, and vertices, where any line segment connecting two points within the shape lies entirely inside it. These shapes are characterized by their convexity, meaning that for any two points within the polyhedron, the line segment connecting them does not exit the polyhedron. Understanding convex polyhedra is essential as they represent a class of geometric objects that frequently appear in optimization problems and can be studied using concepts of convexity and linear programming.
Convex Set: A convex set is a subset of a vector space where, for any two points within the set, the line segment connecting those points also lies entirely within the set. This property makes convex sets fundamental in various mathematical fields, particularly in optimization and geometry, as they simplify many problems and allow for efficient algorithms to operate within their boundaries.
Dual representation: Dual representation refers to a geometric transformation that associates points in a given space with lines in a dual space, providing a way to analyze and visualize geometric properties in different contexts. This concept is fundamental in understanding the relationship between geometric configurations and their duals, often revealing insights into arrangements and the structure of convex sets. In essence, for every geometric element, there's a corresponding dual element, facilitating deeper comprehension of relationships among these objects.
Extreme Points: Extreme points are the vertices or corner points of a convex set, representing the most 'outward' positions that can be reached within that set. These points are crucial for various algorithms in computational geometry, especially when determining the shape and structure of convex hulls. Identifying extreme points allows us to simplify complex geometric problems by focusing only on the boundary of a set rather than its entire area.
Gradient descent: Gradient descent is an optimization algorithm used to minimize a function by iteratively moving toward the steepest descent direction, which is determined by the negative gradient of the function at the current point. This method is particularly important in finding the local minimum of convex functions, where the entire landscape can be navigated effectively due to their unique properties. The efficiency of gradient descent heavily relies on understanding the characteristics of convex sets and their relationship with optimization problems.
Graham's Scan: Graham's Scan is an efficient algorithm used to compute the convex hull of a set of points in the plane. It works by first sorting the points based on their polar angle with respect to a reference point, and then constructing the convex hull by processing these sorted points while maintaining a stack of vertices that form the hull. This method not only establishes the connection between geometric algorithms and convex sets but also showcases practical applications in various fields, such as computer graphics and robotics.
Helly's Theorem: Helly's Theorem states that for a finite collection of convex sets in a Euclidean space, if the intersection of every subset of size at most $d+1$ is non-empty, then the intersection of all the sets is also non-empty, where $d$ is the dimension of the space. This theorem connects deeply to concepts like convexity, helping to determine relationships between sets, and plays a role in understanding Minkowski sums and approximating convex hulls in computational geometry.
Intersection of Convex Sets: The intersection of convex sets refers to the common elements shared between two or more convex sets, resulting in another set that is also convex. This property is crucial because it emphasizes how the structural characteristics of convexity are preserved when combining sets, ensuring that the intersection retains the linear combination property that defines convex sets.
Jarvis March: Jarvis March, also known as the Gift Wrapping algorithm, is a geometric algorithm used to compute the convex hull of a set of points in the plane. This algorithm works by wrapping a 'string' around the outermost points, effectively identifying the boundary of the convex shape formed by those points. It connects to various concepts like the Minkowski sum and applications of convex hulls, highlighting its significance in computational geometry.
Line segment property: The line segment property states that for any two points in a Euclidean space, the shortest distance between them is represented by a straight line segment connecting those points. This concept is fundamental in understanding the structure of convex sets, as it implies that any two points within a convex set can be joined by a line segment that lies entirely within that set.
Linear programming: Linear programming is a mathematical method used for optimizing a linear objective function, subject to linear equality and inequality constraints. It is extensively utilized in various fields to find the best possible outcome, whether maximizing profits or minimizing costs. The relationships between variables in linear programming are represented as convex sets, allowing for efficient solutions through geometrical interpretation, especially in contexts involving facility location and optimization problems.
Optimization Problems: Optimization problems are mathematical challenges that seek to find the best solution from a set of feasible options, often aiming to maximize or minimize a particular objective function. These problems are heavily influenced by the properties of convexity and convex sets, as many optimization techniques utilize these concepts to identify optimal solutions efficiently. In the context of convex sets, an optimization problem is often defined over a convex region where local optima are also global optima, simplifying the solution process.
Point-in-polygon tests: Point-in-polygon tests are computational techniques used to determine whether a specific point lies inside, outside, or on the boundary of a polygon. These tests are essential in various applications like computer graphics, geographic information systems, and robotics, where understanding spatial relationships is crucial. The accuracy and efficiency of these tests can vary depending on the nature of the polygon, such as whether it is convex or concave.
Polar Duality: Polar duality is a concept in computational geometry that describes a relationship between points and lines in the plane, where each point can be associated with a line and each line with a point. This dual relationship allows geometric properties to be analyzed and understood from two perspectives, enhancing the study of convexity and convex sets by revealing insights into their structure and behavior.
Quickhull algorithm: The quickhull algorithm is a computational geometry method used to find the convex hull of a set of points in two-dimensional space. This algorithm operates by recursively dividing the point set into smaller subsets and identifying the extreme points that form the boundary, ultimately resulting in a polygon that encapsulates all other points. Its efficiency makes it particularly valuable for applications requiring quick computations of convex hulls, as well as for understanding the properties of convexity and convex sets.
Separating Axis Theorem: The Separating Axis Theorem states that two convex shapes do not overlap if and only if there exists a line (axis) on which the projections of the two shapes are separated. This theorem is crucial for determining collisions in computational geometry, especially when dealing with complex shapes. It relies on the properties of convex sets, as each edge of the convex shapes can serve as a potential axis for testing separation.
Separation Theorem: The separation theorem is a fundamental concept in convex geometry that states that if two convex sets do not intersect, then there exists a hyperplane that can separate them. This theorem establishes the relationship between convexity and the ability to distinguish between different sets, showing that for any two disjoint convex sets, you can find a line (or hyperplane in higher dimensions) that divides them such that one set lies entirely on one side of the line and the other set on the opposite side.
Support Vector Machines: Support Vector Machines (SVM) are supervised learning models used for classification and regression tasks that aim to find the optimal hyperplane that separates different classes in a dataset. This hyperplane is chosen to maximize the margin between the nearest points of the different classes, known as support vectors, which leads to better generalization on unseen data. The geometric properties of SVM are deeply tied to concepts of convexity and convex sets, as the optimization process relies on the existence of a convex hull formed by the support vectors.
Supporting Hyperplane: A supporting hyperplane is a flat affine subspace of one dimension less than the space it resides in, which intersects a convex set and contains at least one boundary point of that set. It acts as a boundary that separates the convex set from the rest of the space, serving to highlight the concept of convexity by demonstrating how a hyperplane can effectively define the limits of a convex set. Understanding supporting hyperplanes is crucial for analyzing convex sets, as they play a key role in optimization problems and defining the shape of these sets.
Union of Convex Sets: The union of convex sets refers to the combination of two or more convex sets into a single set, where all points from the individual sets are included. While each set maintains its convexity, the union itself may not be convex. This concept highlights the relationship between individual convex shapes and their combined structure, often leading to applications in optimization and geometric algorithms.
Voronoi Diagrams: Voronoi diagrams are a way to partition a plane into regions based on the distance to a specific set of points, known as sites. Each region corresponds to a site and contains all the points that are closer to that site than to any other. This concept is fundamental in understanding spatial structures and is closely related to arrangements of lines and the properties of convex sets.
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