Straszewicz's theorem

Remarks

Proof idea: for a point $z \in \R^n$, a point $p$ of $K$ farthest from $z$ is exposed. Indeed, the sphere around $z$ through $p$ encloses $K$, and its tangent hyperplane at $p$ meets $K$ only in $p$. One then shows that every extreme point can be approached by such farthest points, by choosing $z$ suitably far away in suitable directions.

Combined with the Minkowski–Carathéodory theorem, it follows that $K$ is the convex hull of the closure of its exposed points.

Example: let $K$ be the stadium, the convex hull of the discs of radius $1$ centred at $(-1, 0)$ and $(1, 0)$. The four points $(\pm 1, \pm 1)$ are extreme but not exposed. Each of them is a limit of points on the round ends, and those points are exposed.

Going deeper

For a student

The theorem lets one replace extreme points by exposed points in many arguments, at the cost of a closure. Exposed points are often easier to handle, since each comes with a linear function maximised exactly there. It is used in the study of faces of convex bodies and in convex analysis, where exposed points correspond to directions in which the support function is differentiable.