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]