Abstract
Let H = x ∈ ℝn | p · x = α be a hyperplane in ℝn (p ∈ ℝn \ {0}, α ∈ ℝ). The sets F = x ∈ ℝn | p · x ≤ α and G = x ∈ ℝn | p · x ≥ α are called closed half-spaces associated with (or determined by) H. Obviously, they are convex and topologically closed as the inverse images of the closed intervals of ℝ: ] - ∞, α] and [α,+ ∞[ respectively by the (continuous) linear functional: x → p · x. The complementary sets V = x ∈ℝn | p · x > α and U = x ∈ℝn | p · x < α are obviously convex and topologically open; they are called open half-spaces associated with (or determined by) H.
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© 2001 Springer-Verlag Berlin Heidelberg
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Florenzano, M., Le Van, C. (2001). Separation and Polarity. In: Finite Dimensional Convexity and Optimization. Studies in Economic Theory, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56522-9_2
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DOI: https://doi.org/10.1007/978-3-642-56522-9_2
Publisher Name: Springer, Berlin, Heidelberg
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