0. Convexity

Some subsets of R$^2$ are convex, and others are not, as in Figure .

Figure: Convex and nonconvex sets

It is important to observe that convexity is a property of the whole set, and not just of its boundary. For example, the circle $x^2 + y^2 = 1$ in R$^2$ is not convex, but the circular disk $x^2 + y^2 \leq 1$ is convex.

Convexity can be tested by looking at line segments with both end points in the set. In fact, this test makes a good official definition, in any number of dimensions:

Definition. A subset $C$ of R$^n$ is convex if for every two points $P,Q$ in $C$, the whole line segment $\overline {PQ}$ is in $C$.

In R$^3$, for example, a solid cube is convex, a tetrahedron is convex, and a ball (solid sphere) is convex, but a torus is not and a (hollow) sphere is not.