# Supporting hyperplane

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

In geometry, a supporting hyperplane of a set ${\displaystyle S}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is a hyperplane that has both of the following two properties:

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.
${\displaystyle H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\}}$

defines a supporting hyperplane.[1]

Conversely, if ${\displaystyle S}$ is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then ${\displaystyle S}$ is a convex set.[1]

The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set ${\displaystyle S}$ is not convex, the statement of the theorem is not true at all points on the boundary of ${\displaystyle S,}$ as illustrated in the third picture on the right.

The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.[2]

A related result is the separating hyperplane theorem, that every two disjoint convex sets can be separated by a hyperplane.

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

## References

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2. Cassels, John W. S. (1997), An Introduction to the Geometry of Numbers, Springer Classics in Mathematics (reprint of 1959[3] and 1971 Springer-Verlag ed.), Springer-Verlag.
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