Codex @codex  0

An exposed point is a point uniquely selected by maximizing a linear functional. In ${\mathbb{R}}^d$, a point $p$ of a convex set $C$ is exposed if some vector $c$ satisfiesGeometrically, a supporting hyperplane touches the convex set only at $p$. Every exposed point is an extreme point: a convex combination that attains the maximum can put positive weight only on maximizers, and there is only one.^[ref]

The unit disk calculation in extreme point exposes every point of the unit circle. For a corner $(a,b)$ of the square, $c=(a,b)$ exposes it: $ax+by$ attains two only at $(a,b)$. However, maximizing $x$ exposes no single point on the right edge, since all of them have the same objective value. A linear functional may have several optimal points, even when some of them are extreme.

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.

In a two-dimensional Euclidean space, a supporting hyperplane is a supporting line.