# Supporting hyperplane

*of a set $M$ in an $n$-dimensional vector space*

An $(n-1)$-dimensional plane containing points of the closure of $M$ and leaving $M$ in one closed half-space. When $n=3$, a supporting hyperplane is called a supporting plane, while when $n=2$, it is called a supporting line.

A boundary point of $M$ through which at least one supporting hyperplane passes is called a support point of $M$. In a convex set $M$, all boundary points are support points. This property was used by Archimedes as a definition of the convexity of $M$. Boundary points of a convex set $M$ through which only one supporting hyperplane passes are called smooth.

In general vector spaces, where a hyperplane can be defined as a domain of constant value of a linear functional, the concept of a supporting hyperplane of a set $M$ can also be defined (the values of the linear functional at the points of $M$ should be all less (all greater) than or equal to the value the linear functional takes on the hyperplane).

#### Comments

Supporting hyperplanes are also of importance in applications of convexity, e.g. optimization and geometry of numbers.

#### References

[a1] | P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) |

[a2] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307 |

[a3] | R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59 |

[a4] | J. Stoer, C. Witzgall, "Convexity and optimization in finite dimensions" , 1 , Springer (1970) |

[a5] | J. Lindenstrauss (ed.) V.D. Milman (ed.) , Geometric aspects of functional analysis , Lect. notes in math. , 1376 , Springer (1988) |

**How to Cite This Entry:**

Supporting hyperplane.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Supporting_hyperplane&oldid=33306