Support Function

Definition

The support function is a mathematical tool used to describe the boundary behavior of convex sets. It is defined for a convex set as the maximum value of the inner product between a fixed vector and points in the set. This function helps in analyzing various properties of convex sets, including their geometry and relationships with other mathematical concepts such as polar sets and conjugate functions.

5 Must Know Facts For Your Next Test

1. The support function of a convex set $C$ at a point $u$ is defined as $h_C(u) = ext{sup}\{\langle x, u \rangle : x \in C\}$, where $\langle x, u \rangle$ represents the inner product.
2. Support functions can help identify supporting hyperplanes of convex sets, which are crucial for understanding their geometric structure.
3. The support function is positively homogeneous, meaning that $h_C(\alpha u) = \alpha h_C(u)$ for any positive scalar $\alpha$.
4. The support function can be used to define properties of convex cones and hulls, allowing us to examine their structure through functional forms.
5. For any convex set, its support function is subadditive, implying that for two sets $A$ and $B$, we have $h_{A+B}(u) \leq h_A(u) + h_B(u)$.

Review Questions

• How does the support function relate to the properties of convex cones?
  • The support function plays a vital role in understanding convex cones by defining their geometric properties through supporting hyperplanes. In particular, it helps determine how far the cone extends in certain directions and aids in examining its boundaries. By analyzing the support function, one can ascertain key features such as the directionality of the cone and its interaction with other geometric structures.
• Discuss how the support function aids in characterizing convex hulls and their properties.
  • The support function serves as a powerful tool for characterizing convex hulls by providing a way to express their boundary points mathematically. Specifically, it allows for identifying the extreme points that compose the convex hull of a given set. By using the support function, one can derive essential properties such as compactness and connectedness of convex hulls, which are significant in various applications, including optimization and computational geometry.
• Evaluate the significance of the support function in relation to weak topologies and Fenchel duality.
  • The support function's significance in weak topologies stems from its ability to bridge relationships between different mathematical constructs such as functional forms and continuity. In the context of Fenchel duality, the support function provides insights into optimizing problems by allowing comparisons between primal and dual formulations. By analyzing support functions under weak topologies, we can ensure consistency in dual relationships while maintaining properties like compactness and convergence within convex sets.

Related terms

Convex Set: A set is convex if, for any two points within the set, the line segment connecting them also lies entirely within the set.

Polar Set: For a given set, the polar set consists of all vectors that have non-negative inner products with every vector in the original set.

Conjugate Function: The conjugate function is derived from a given function and provides insights into its properties, often relating to optimization and duality.