7.1 Polar sets and their properties

Polar sets are a fundamental concept in convex geometry, offering a dual perspective on convex sets. They provide a way to represent a set through its linear constraints, revealing important geometric and algebraic properties.

Understanding polar sets is crucial for grasping duality in convex analysis. They have practical applications in optimization, helping to reformulate problems and derive dual formulations, which can lead to more efficient solution methods.

Polar Sets and Their Properties

Polar sets in convex geometry

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• Polar set definition
  • For set $S$ in $\mathbb{R}^n$, polar set $S^\circ$ defined as $S^\circ = \{y \in \mathbb{R}^n : \langle x, y \rangle \leq 1 \text{ for all } x \in S\}$
  • $\langle x, y \rangle$ denotes inner product of vectors $x$ and $y$ measures alignment between vectors
• Geometric interpretation
  • Collection of points forming hyperplanes separating origin from $S$ creates boundary between set and origin
  • Represents set of linear functionals bounded by 1 on $S$ captures linear constraints on original set
  • Dual representation of original set provides alternative perspective on set's structure
• Key characteristics
  • Always closed convex set containing origin ensures well-defined geometric properties
  • Generalizes orthogonality concept to sets extends perpendicularity to higher dimensions

Properties of polar sets

• Convexity of polar sets
  • Proof: For $y_1, y_2 \in S^\circ$ and $λ \in [0,1]$, show $λy_1 + (1-λ)y_2 \in S^\circ$ demonstrates closure under convex combinations
  • Application: Simplifies analysis and computations involving polar sets (optimization problems, duality theory)
• Symmetry properties
  • Central symmetry: If $S$ centrally symmetric about origin, $S^\circ$ also centrally symmetric reflects original set's symmetry
  • Proof: If $y \in S^\circ$, then $-y \in S^\circ$ demonstrates reflection property
• Inclusion-reversing property
  • If $A \subseteq B$, then $B^\circ \subseteq A^\circ$ shows inverse relationship between set containment
  • Useful for comparing and analyzing nested convex sets (hierarchical structures, containment problems)
• Bipolar theorem
  • $(S^\circ)^\circ = \overline{\text{conv}}(S \cup \{0\})$ connects original set to its double polar
  • Important for understanding duality in convex geometry reveals fundamental relationship between set and its polar

Polar sets and support functions

• Support function definition
  • For set $S$, support function $h_S$ defined as $h_S(y) = \sup_{x \in S} \langle x, y \rangle$ measures extent of set in direction $y$
• Relationship between support functions and polar sets
  • $y \in S^\circ$ if and only if $h_S(y) \leq 1$ characterizes polar set using support function
  • Polar set as unit ball of support function provides geometric interpretation of support function
• Duality of support functions
  • $h_{S^\circ}(x) = \sup_{y \in S^\circ} \langle x, y \rangle = I_S(x)$ connects support functions of set and its polar
  • $I_S(x)$ is indicator function of $S$ equals 0 for $x \in S$, infinity otherwise
• Applications in optimization
  • Support functions reformulate convex optimization problems (dual problems, Lagrangian relaxation)
  • Polar sets employed in duality theory for linear and convex programming (strong duality, complementary slackness)

Computation of polar sets

• Polar set of a polytope
  • For polytope $P = \text{conv}\{v_1, \ldots, v_k\}$ defined by vertices
  • $P^\circ = \{y : \langle v_i, y \rangle \leq 1 \text{ for all } i = 1, \ldots, k\}$ represents intersection of halfspaces
  • Represents intersection of halfspaces defined by vertices forms faceted structure
• Polar set of an ellipsoid
  • For ellipsoid $E = \{x : x^T A x \leq 1\}$ where $A$ positive definite
  • $E^\circ = \{y : y^T A^{-1} y \leq 1\}$ inverts matrix $A$
  • Demonstrates duality between ellipsoids reveals inverse relationship in shape
• Polar set of a ball
  • For ball $B_r = \{x : \|x\| \leq r\}$ with radius $r$
  • $B_r^\circ = \{y : \|y\| \leq \frac{1}{r}\}$ inverts radius
  • Illustrates inverse relationship between radii of set and its polar shows scaling property
• Computational techniques
  • Vertex enumeration for polytopes identifies extreme points
  • Matrix operations for ellipsoids involves matrix inversion
  • Geometric transformations for complex sets applies affine transformations