 Supporting hyperplane

For a convex set $C$ and $x_0\in C$, a supporting hyperplane has a nonzero normal vector $a$ such that $a^{T}x\le a^T{x}_0$ for all $x\in C$. It touches $C$ while containing the set on one side.

A **supporting hyperplane** is a concept from convex analysis and geometry, particularly in the context of convex sets and optimization. It relates to how we can visualize and understand the boundaries of convex sets in multidimensional spaces. Formally, a hyperplane can be defined as a flat, affine subspace of one dimension less than the dimension of the surrounding space. For example, in a 3-dimensional space, a hyperplane is a 2-dimensional plane.