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

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

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} 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,

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

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

Let b = t ( y − x ) + x {\displaystyle b=t(y-x)+x}, then b ∈ ∂ S {\displaystyle b\in \partial S}. Draw a supporting hyperplane across b {\displaystyle b}. Let it be represented as a nonzero linear functional f : R n → 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)}. Then since x ∈ i n t ( S ) {\displaystyle x\in \mathrm {int} (S)}, we have 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}, we have f ( y ) < f ( b ) {\displaystyle f(y)<f(b)}, so y ∉ T {\displaystyle y\not \in T}.

See also

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

Notes

References & further reading

  • Ostaszewski, Adam (1990). . Cambridge; New York: Cambridge University Press. p. . 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.