In geometry, a supporting hyperplane of a set S {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both of the following two properties:
S {\displaystyle S} is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
S {\displaystyle S} has at least one boundary-point on the hyperplane. Here, a closed half-space is the half-space that includes the points within the hyperplane.
Supporting hyperplane theorem
This theorem states that if S {\displaystyle S} is a convex set in the topological vector space X = R n , {\displaystyle X=\mathbb {R} ^{n},} and x 0 {\displaystyle x_{0}} is a point on the boundary of S , {\displaystyle S,} then there exists a supporting hyperplane containing x 0 . {\displaystyle x_{0}.} If x ∗ ∈ X ∗ ∖ { 0 } {\displaystyle x^{*}\in X^{*}\backslash \{0\}} ( X ∗ {\displaystyle X^{*}} is the dual space of X {\displaystyle X} , x ∗ {\displaystyle x^{*}} is a nonzero linear functional) such that x ∗ ( x 0 ) ≥ x ∗ ( x ) {\displaystyle x^{*}\left(x_{0}\right)\geq x^{*}(x)} for all x ∈ S {\displaystyle x\in S} , then
H = { x ∈ X : x ∗ ( x ) = x ∗ ( x 0 ) } {\displaystyle H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\}}
defines a supporting hyperplane. Conversely, if S {\displaystyle S} is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then S {\displaystyle S} is a convex set, and is the intersection of all its supporting closed half-spaces. The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set S {\displaystyle S} is not convex, the statement of the theorem is not true at all points on the boundary of S , {\displaystyle S,} as illustrated in the third picture on the right. The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes. The forward direction can be proved as a special case of the separating hyperplane theorem (see the page for the proof). For the converse direction,
See also
Support function Supporting line (supporting hyperplanes in R 2 {\displaystyle \mathbb {R} ^{2}} )
Notes
References & further reading Ostaszewski, Adam (1990). Advanced mathematical methods, London School of Economics Mathematics Series. Cambridge; New York: Cambridge University Press. p. 129. ISBN 0-521-28964-5. Giaquinta, Mariano; Hildebrandt, Stefan (1996). Calculus of variations. Berlin; New York: Springer. p. 57. ISBN 3-540-50625-X. Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. p. 13. ISBN 0-415-27479-6. Soltan, V. (2021). Support and separation properties of convex sets in finite dimension. Extracta Math. Vol. 36, no. 2, 241-278.
