A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem

Ali Taherinassaj; Yiling Chen

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arXiv · 2303.05423

A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem

Ali Taherinassaj·Yiling Chen

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Abstract

We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The insight is that two sets are separable by a hyperplane if there is a point from whose perspective the two sets appear to be convex. This global topological to local vector space relationship anticipates generalizations to manifolds as a possible future direction. The result might also be of pedagogical interest as it re-wires the structure in which the supporting hyperplane theorem and its dependencies are usually presented in convex optimization and analysis textbooks.

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Ali Taherinassaj, Yiling Chen. 2023-03-06. A Non-Convex Separation Through an Alternative Proof for the Supporting Hyperplane Theorem. https://arxiv.org/abs/2303.05423

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