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[edit]

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

#### References[edit]

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

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[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)