Supporting hyperplane

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:^[1]

• ${\displaystyle S}$ is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
• ${\displaystyle S}$ has at least one boundary-point on the hyperplane.

Here, a closed half-space is the half-space that includes the points within the hyperplane.