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Supporting hyperplane

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Hyperplane in geometry
A convex set S {\displaystyle S} {\displaystyle S} (in pink), a supporting hyperplane of S {\displaystyle S} {\displaystyle S} (the dashed line), and the supporting half-space delimited by the hyperplane which contains S {\displaystyle S} {\displaystyle S} (in light blue).

In geometry, a supporting hyperplane of a set S {\displaystyle S} {\displaystyle S} in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}} is a hyperplane that has both of the following two properties:[1]

  • S {\displaystyle S} {\displaystyle S} is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
  • S {\displaystyle 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

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A convex set can have more than one supporting hyperplane at a given point on its boundary.

This theorem states that if S {\displaystyle S} {\displaystyle S} is a convex set in the topological vector space X = R n , {\displaystyle X=\mathbb {R} ^{n},} {\displaystyle X=\mathbb {R} ^{n},} and x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} is a point on the boundary of S , {\displaystyle S,} {\displaystyle S,} then there exists a supporting hyperplane containing x 0 . {\displaystyle x_{0}.} {\displaystyle x_{0}.} If x ∗ ∈ X ∗ ∖ { 0 } {\displaystyle x^{*}\in X^{*}\backslash \{0\}} {\displaystyle x^{*}\in X^{*}\backslash \{0\}} ( X ∗ {\displaystyle X^{*}} {\displaystyle X^{*}} is the dual space of X {\displaystyle X} {\displaystyle X}, x ∗ {\displaystyle x^{*}} {\displaystyle x^{*}} is a nonzero linear functional) such that x ∗ ( x 0 ) ≥ x ∗ ( x ) {\displaystyle x^{*}\left(x_{0}\right)\geq x^{*}(x)} {\displaystyle x^{*}\left(x_{0}\right)\geq x^{*}(x)} for all x ∈ S {\displaystyle x\in 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)\}} {\displaystyle H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\}}

defines a supporting hyperplane.[2]

Conversely, if S {\displaystyle 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} {\displaystyle S} is a convex set, and is the intersection of all its supporting closed half-spaces.[2]

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} {\displaystyle S} is not convex, the statement of the theorem is not true at all points on the boundary of S , {\displaystyle 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.[3]

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,

Proof

Define T {\displaystyle T} {\displaystyle T} to be the intersection of all its supporting closed half-spaces. Clearly S ⊂ T {\displaystyle S\subset T} {\displaystyle S\subset T}. Now let y ∉ S {\displaystyle y\not \in S} {\displaystyle y\not \in S}, show y ∉ T {\displaystyle y\not \in T} {\displaystyle y\not \in T}.

Let x ∈ i n t ( S ) {\displaystyle x\in \mathrm {int} (S)} {\displaystyle x\in \mathrm {int} (S)}, and consider the line segment [ x , y ] {\displaystyle [x,y]} {\displaystyle [x,y]}. Let t {\displaystyle t} {\displaystyle t} be the largest number such that [ x , t ( y − x ) + x ] {\displaystyle [x,t(y-x)+x]} {\displaystyle [x,t(y-x)+x]} is contained in S {\displaystyle S} {\displaystyle S}. Then t ∈ ( 0 , 1 ) {\displaystyle t\in (0,1)} {\displaystyle t\in (0,1)}.

Let b = t ( y − x ) + x {\displaystyle b=t(y-x)+x} {\displaystyle b=t(y-x)+x}, then b ∈ ∂ S {\displaystyle b\in \partial S} {\displaystyle b\in \partial S}. Draw a supporting hyperplane across b {\displaystyle b} {\displaystyle b}. Let it be represented as a nonzero linear functional f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } such that ∀ a ∈ T , f ( a ) ≥ f ( b ) {\displaystyle \forall a\in T,f(a)\geq f(b)} {\displaystyle \forall a\in T,f(a)\geq f(b)}. Then since x ∈ i n t ( S ) {\displaystyle x\in \mathrm {int} (S)} {\displaystyle x\in \mathrm {int} (S)}, we have f ( x ) > f ( b ) {\displaystyle f(x)>f(b)} {\displaystyle f(x)>f(b)}. Thus by f ( y ) − f ( b ) 1 − t = f ( b ) − f ( x ) t − 0 < 0 {\displaystyle {\frac {f(y)-f(b)}{1-t}}={\frac {f(b)-f(x)}{t-0}}<0} {\displaystyle {\frac {f(y)-f(b)}{1-t}}={\frac {f(b)-f(x)}{t-0}}<0}, we have f ( y ) < f ( b ) {\displaystyle f(y)<f(b)} {\displaystyle f(y)<f(b)}, so y ∉ T {\displaystyle y\not \in T} {\displaystyle y\not \in T}.

See also

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A supporting hyperplane containing a given point on the boundary of S {\displaystyle S} {\displaystyle S} may not exist if S {\displaystyle S} {\displaystyle S} is not convex.

Notes

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  1. ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 133. ISBN 978-0-471-18117-0.
  2. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 50–51. ISBN 978-0-521-83378-3 . Retrieved October 15, 2011.
  3. ^ Cassels, John W. S. (1997), An Introduction to the Geometry of Numbers, Springer Classics in Mathematics (reprint of 1959[3] and 1971 Springer-Verlag ed.), Springer-Verlag.

References & further reading

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  • 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.

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