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Supporting hyperplane

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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://www.encyclopediaofmath.org/index.php?title=Supporting_hyperplane&oldid=33306
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article