1.2 Examples and counterexamples of convex sets
3 min read•july 25, 2024
Convex sets are fundamental in geometry, forming the building blocks of many mathematical structures. From simple line segments to complex higher-dimensional shapes, these sets share the property that any line connecting two points within the set lies entirely inside it.
Non-convex sets, on the other hand, break this rule. They include shapes with indentations, holes, or disconnected parts. Understanding the difference between convex and non-convex sets is crucial for analyzing geometric properties and solving optimization problems in various fields.
Examples of Convex Sets
Examples of convex sets
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- One-dimensional convex sets
- Line segments connect two points with all points between them included
- Rays extend infinitely in one direction from a starting point
- Real number line encompasses all real numbers on a continuous line
- Two-dimensional convex sets
- Circles and discs contain all points within a given distance from the center
- Triangles enclose area with three straight sides and three angles
- Rectangles and squares have four right angles and parallel sides
- Regular polygons feature equal sides and angles (pentagons, hexagons)
- Half-planes divide the plane into two regions by a straight line
- Three-dimensional convex sets
- Spheres and balls include all points within a certain distance from center point
- Cubes and cuboids have six rectangular faces at right angles
- Cylinders consist of two parallel circular bases connected by a curved surface
- Cones have circular base tapering to a point (apex)
- Tetrahedra formed by four triangular faces meeting at four vertices
- Higher-dimensional convex sets
- Hypercubes generalize cubes to higher dimensions (tesseracts in 4D)
- Simplices extend triangles and tetrahedra to n-dimensions
- Ellipsoids generalize circles and spheres with stretched axes
- Hyperplanes and half-spaces divide n-dimensional space
Counterexamples for non-convex sets
- Two-dimensional non-convex sets
- Crescent shapes formed by intersecting two circles
- Star-shaped polygons with points extending outward (pentagrams)
- Annuli create ring-shaped regions between two concentric circles
- Three-dimensional non-convex sets
- Torus resembles a donut with a hole through the center
- Hollow spheres have empty interior with only surface points
- L-shaped solids formed by joining two rectangular prisms
- Sets with disconnected components
- of disjoint circles creates separate circular regions
- Set of integers on the real line forms discrete, separated points
- Non-convex regions defined by inequalities
- Set satisfying creates area outside unit circle
Analysis of Convexity
Convexity of geometric shapes
- Convex polygons
- All interior angles measure less than 180 degrees
- Every between two points remains inside polygon
- Non-convex polygons
- At least one interior angle exceeds 180 degrees
- Some line segments between points fall outside polygon
- Curved shapes
- Ellipses remain convex with smooth, continuous curvature
- Parabolas maintain convexity with symmetric, U-shaped curve
- Hyperbolas form non-convex shape with two separate branches
- Polyhedra
- Convex polyhedra contain only convex polygonal faces (cubes, tetrahedra)
- Non-convex polyhedra exhibit indentations or holes (stellated dodecahedron)
Convexity in inequality-defined sets
- Linear inequalities
- Single linear inequality creates half-space (convex)
- System of linear inequalities forms convex polytope (, cube)
- Quadratic inequalities
- defines circular disc (convex)
- produces hyperbolic region (non-convex)
- Norm-based inequalities
- generates ball in any norm (convex)
- creates (complement of unit ball)
- of convex sets
- Always yields due to preservation of convexity
- Intersection of half-spaces produces convex polyhedron
- Union of convex sets
- Generally results in non-convex set (union of two separate circles)
- Exception occurs when one set fully contains the other
Key Terms to Review (17)
Bergström: Bergström refers to a specific geometric construction and concept within the realm of convex geometry that deals with the analysis of convex sets. This term is often associated with the characterization and classification of different types of convex shapes, as well as providing examples that illustrate their properties. Understanding Bergström is crucial for identifying the nuances between various convex sets and recognizing counterexamples that showcase deviations from convexity.
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 highlights the relationship between points and their extreme points, connecting to the geometric understanding of convex sets and their properties.
Concave Polygon: A concave polygon is a type of polygon where at least one of its interior angles is greater than 180 degrees, resulting in at least one vertex pointing inward toward the interior of the shape. This characteristic distinguishes it from convex polygons, where all interior angles are less than 180 degrees and all vertices point outward. Understanding concave polygons is crucial for identifying and contrasting various geometric shapes in the study of convex sets.
Convex envelope: The convex envelope of a set of points is the smallest convex set that contains all the points. It can be thought of as the 'tightest' convex shape that can wrap around a given set of points, ensuring that any line segment connecting two points in the set lies entirely within the convex envelope. This concept is crucial for understanding various applications in geometry and statistical learning, as it helps to characterize the structure and properties of both convex sets and data distributions.
Convex Hull: The convex hull of a set of points is the smallest convex set that contains all the points. It can be visualized as the shape formed by stretching a rubber band around the outermost points, effectively enclosing them in the tightest possible way.
Convex polygon: A convex polygon is a simple polygon in which all interior angles are less than 180 degrees, and for any two points within the polygon, the line segment connecting them lies entirely within the polygon. This property ensures that the polygon has a 'bulging' outward shape without any indentations or 'dents' in its edges. Convex polygons are a fundamental concept in convex geometry, distinguishing them from concave polygons where at least one interior angle exceeds 180 degrees.
Convex Set: A convex set is a subset of a vector space where, for any two points within the set, the line segment connecting them lies entirely within the set. This property ensures that any combination of points in the set can be reached without leaving it, making convex sets fundamental in various mathematical contexts, such as optimization and geometry.
Extreme Point: An extreme point of a convex set is a point that cannot be expressed as a convex combination of other points in the set. These points are crucial in understanding the shape and boundaries of convex sets, playing a significant role in optimization problems and the structure of convex functions.
Intersection: In geometry, the intersection refers to the set of points that are common to two or more sets. Understanding intersections is crucial because they illustrate how different geometric figures can relate to one another, especially when considering properties of convex sets and their separations.
Line Segment: A line segment is a part of a line that is bounded by two distinct endpoints. It is the simplest form of a geometric object, connecting two points in space and representing the shortest distance between them. In the context of convex sets, line segments play a crucial role in determining whether a set is convex, as any two points within a convex set can be connected by a line segment that lies entirely within that set.
Non-convex set: A non-convex set is a type of set in geometry where at least two points within the set can be connected by a line segment that lies partially or entirely outside of the set. This lack of convexity means that non-convex sets can have 'dents' or 'holes,' making them distinctly different from convex sets, where any two points can be connected without leaving the set. Understanding non-convex sets is crucial when discussing various geometric properties and their implications in mathematics.
Separation Theorem: The separation theorem states that for two disjoint convex sets, there exists a hyperplane that can separate them in such a way that one set lies entirely on one side of the hyperplane and the other set lies entirely on the opposite side. This theorem plays a crucial role in understanding the geometric relationships between convex sets and is foundational for many results in optimization, geometry, and functional analysis.
Simplex: A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions, serving as the simplest type of convex polytope. In a $d$-dimensional space, a simplex is defined by $d + 1$ vertices, which are not all contained in any single hyperplane. This concept is foundational for understanding convex sets, polyhedra, and the structure of polytopes, as it lays the groundwork for more complex geometric objects.
Star-shaped set: A star-shaped set is a subset of a vector space where, for every point within the set, the line segment connecting that point to a specific point (the star center) in the set remains entirely within the set. This property means that if you pick any point in the star-shaped set, you can always draw a straight line to the star center without leaving the set. Understanding this concept is crucial when distinguishing star-shaped sets from convex sets, as all convex sets are star-shaped, but not all star-shaped sets are convex.
Supporting Hyperplane: A supporting hyperplane is a flat affine subspace of one dimension less than the ambient space that touches a convex set at least at one point, and such that the convex set lies entirely on one side of the hyperplane. This concept is crucial in understanding how convex sets interact with linear functions and is foundational in various applications, including optimization and geometry.
Tikhonov: Tikhonov refers to a concept in convex analysis, specifically related to Tikhonov regularization, which is a method used in mathematical optimization and functional analysis. This approach aims to stabilize ill-posed problems by adding a regularization term, leading to solutions that have better numerical properties and are more robust to noise. Understanding Tikhonov is essential when exploring examples and counterexamples of convex sets, as it provides insight into how convexity interacts with optimization techniques.
Union: In set theory, the union of two or more sets is a set that contains all the elements from the involved sets without duplication. This concept is crucial in understanding how different geometric shapes can interact, especially when determining if a combination of convex sets still retains the properties of convexity.