3.3 Supporting hyperplanes and their properties

Supporting hyperplanes are crucial in convex geometry. They touch convex sets at boundary points without intersecting the interior, acting like tangent planes. These hyperplanes divide space into two half-spaces, with the convex set lying entirely on one side.

Supporting hyperplanes have key properties like separation, tangency, and non-intersection with the interior. They exist at every boundary point of closed convex sets, with smooth points having unique hyperplanes and non-smooth points potentially having multiple. This concept is closely tied to normal cones in convex analysis.

Supporting Hyperplanes and Their Properties

Supporting hyperplanes for convex sets

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• Hyperplane divides space into two half-spaces affine subspace one dimension less than ambient space
• Supporting hyperplane touches convex set at boundary points without intersecting interior (tangent plane)
• Mathematical definition: For convex set $C$ in $\mathbb{R}^n$, hyperplane $H$ supporting if $H \cap C \neq \emptyset$ and $C$ contained in one closed half-space defined by $H$
• Algebraic representation: $H = \{x \in \mathbb{R}^n : a^Tx = b\}$ with $a$ non-zero normal vector and $b$ scalar (plane equation)

Properties of supporting hyperplanes

• Separation property keeps convex set apart from external points (separating barrier)
• Tangency occurs at one or more boundary points (touching without crossing)
• Non-intersection with interior maintains convex set integrity (external contact only)
• Orientation normal vector points outward from convex set (directional indicator)
• Half-space containment ensures convex set lies entirely on one side (spatial restriction)
• Local support approximates convex set near contact point (local linearization)

Existence at convex set boundaries

• Existence theorem guarantees supporting hyperplane at every boundary point of closed convex set
• Non-smooth points may have multiple supporting hyperplanes (corner points)
• Smooth points have unique supporting hyperplane (differentiable surface)
• Finite-dimensional spaces always have supporting hyperplanes for closed convex sets
• Infinite-dimensional spaces may require additional conditions for existence
• Construction methods include separating hyperplane theorem and subgradients of convex functions

Relation to normal cones

• Normal cone comprises all outward-pointing normal vectors to supporting hyperplanes at a point
• Each vector in normal cone defines a supporting hyperplane (geometric correspondence)
• Mathematical representation: $N_C(x) = \{v : v^T(y-x) \leq 0, \forall y \in C\}$ for convex set $C$ and point $x \in C$
• Properties include convexity, closure, and containing zero vector
• Smooth boundary points have ray-like normal cone (single outward direction)
• Non-smooth points may have higher-dimensional normal cone (multiple outward directions)
• Applications in convex optimization optimality conditions and convex set projection characterization