# 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).