The Supporting Hyperplane Theorem is a fundamental result in convex analysis, particularly important in optimization and economics. It states that for any convex set in a vector space, at any boundary point of the set, there exists a hyperplane that touches the set at that point and does not intersect the interior of the set.

Statement of the Theorem

Let $C$ be a non-empty convex subset of $\mathbb{R}^n$ and let $x_0$ be a boundary point of $C$. Then, there exists a non-zero vector $a \in \mathbb{R}^n$ and a real number $b$ such that: $a \cdot x \leq b \quad \text{for all } x \in C$ and $a \cdot x_0 = b.$

Proof Outline

1. Convexity and Boundary Point:
   Since $C$ is convex and $x_0$ is a boundary point, there exists a point $y \notin C$ such that the line segment between $x_0$ and $y$ crosses the boundary of $C$.

2. Defining the Functional: Consider the functional $f(x) = \lambda a \cdot x + b$ where $\lambda$ is a scalar and $a$ is a non-zero vector. We aim to define $a$ and $b$ such that this functional represents a hyperplane that supports $C$ at $x_0$.

3. Separation of Points: Since $y \notin C$ and $x_0$ is on the boundary, we can use the Hahn-Banach Separation Theorem (a more general result in functional analysis). This theorem guarantees that there exists a linear functional $a \cdot x$ and a scalar $\alpha$ such that: $a \cdot x_0 = \alpha \quad \text{and} \quad a \cdot y > \alpha.$ The condition $a \cdot y > \alpha$ ensures that $y$ is outside the convex set $C$, while $a \cdot x_0 = \alpha$ implies that the hyperplane passes through $x_0$.

4. Normalization: Without loss of generality, normalize $a$ so that $b = \alpha$, and rewrite the inequalities: $a \cdot x_0 = b \quad \text{and} \quad a \cdot x \leq b \quad \text{for all } x \in C.$

5. Conclusion: The linear functional $a \cdot x \leq b$ defines a hyperplane that supports the convex set $C$ at $x_0$. The set $C$ lies entirely on one side of this hyperplane, touching it exactly at $x_0$.

Detailed Version of the Proof:

If you would like to delve into the details of each step, including the use of the Hahn-Banach theorem and further implications in different contexts, I can expand on each section.

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Would you like further details on this proof or have any specific questions? Here are some related questions you might be interested in:

1. How does the Hahn-Banach Separation Theorem work in this context?
2. Can this theorem be extended to non-Euclidean spaces?
3. What are some applications of the Supporting Hyperplane Theorem in optimization?
4. How does the theorem relate to convex functions?
5. What is the role of the boundary point $x_0$ in this theorem?

Tip: Understanding the Supporting Hyperplane Theorem can significantly aid in grasping more complex optimization problems, particularly in linear programming.