1.1 Definition and basic properties of convex sets

Convex sets form the foundation of convex geometry. They're defined by a simple rule: any line segment between two points in the set must be entirely contained within it. This property leads to powerful mathematical tools and applications.

Convex sets have unique properties that make them useful in optimization and analysis. They can be scaled, translated, and intersected while maintaining convexity. Understanding these properties helps solve complex problems in various fields.

Understanding Convex Sets

Definition of convex sets

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• Convex set defined as subset C of vector space where line segment between any two points x and y in C entirely contained within C
• Mathematically represented as $λx + (1-λ)y ∈ C$ for any x, y ∈ C and λ ∈ [0, 1]
• Closed under convex combinations preserving shape integrity
• Intersection of convex sets remains convex maintaining common properties
• Union of convex sets may not be convex potentially creating non-convex regions
• Common convex sets include line segments, hyperplanes, half-spaces, balls, spheres, polygons, polyhedra

Properties of convex sets

• Convexity preserved under scaling $αC$ for any scalar α
• Translation $C + v$ maintains convexity for any vector v
• Intersection $\cap_{i=1}^n C_i$ convex if each $C_i$ convex
• Separating hyperplane theorem states disjoint convex sets can be separated by hyperplane
• Supporting hyperplane theorem ensures existence of supporting hyperplane at boundary points of closed convex set
• Caratheodory's theorem expresses points in convex hull as combination of at most n+1 points in n-dimensional space
• Jensen's inequality states $f(\sum_{i=1}^n λ_i x_i) ≤ \sum_{i=1}^n λ_i f(x_i)$ for convex function f and convex combination of points

Geometric interpretation of convexity

• Convex sets visually "bulge outwards" without indentations or holes
• Convex hull forms smallest convex set containing given points (rubber band analogy)
• Extreme points cannot be expressed as convex combination of other points (polygon vertices)
• Facial structure includes faces, edges, vertices of convex polyhedra
• Dimensionality distinguishes full-dimensional from lower-dimensional convex sets

Convex vs non-convex sets

• Non-convex sets characterized by "dents" or "holes" (tori, star-shaped polygons)
• Line segments between some points in non-convex sets lie outside the set
• Convexity tested through midpoint convexity or convex combination methods
• Convexification achieved by taking convex hull or applying relaxation techniques
• Convex problems generally easier to solve in optimization
• Non-convex problems may have multiple local optima complicating solution process