Convex polygon :   

A polygon is said convex if the line that joins any two points of the polygon is entirely inside the polygon.

Any diagonal will be inside the polygon
Convex polygon
Some diagonals are outside of the polygon
Non-convex polygon

Metric :

A function D, usually called distance, that respects the following properties when applied to x, y and z :

1.  D(x, y) >= 0 (positiveness)
2.  D(x, y) = 0 iff x = y (identity)
3.  D(x, y) = D(y, x) (symmetry)
4.  D(x, y) + D(y, z) >= D(x, z) (triangle inequality)

Monotone chain :

A chain C (i.e. a sequence of edges) is said to be monotone in a direction D if any line L orthogonal to D intersects C in exactly one point.


Never more than one intersection with L
Monotone chain
More than one intersection with L
Non-monotone chain

A polygon P is monotone in some direction if an orthogonal line intersects P in no more than two points ;  a convex polygon is monotone in any direction.

Simple polygon :

A polygon with no self-intersecting edges. Since a polygon is defined by a sequence of vertices, a priori nothing prevents any self-intersection of its edges.

No edges intersect

Simple polygon
Self-intersecting edges

Non-simple polygon

Supporting line :

Straight line L passing through a vertex of a polygon P, such that the interior of P lies entirely on one side of L.  The supporting line is a generalization of the tangent.

L is a supporting line