What is a lower convex hull? (Of a finite number of points in the plane)

Question

I am trying to understand Newton Polygons for computing Galois Groups and they are defined as a lower convex hull (see the third page of http://people.math.gatech.edu/~mbaker/pdf/Coleman_GaloisNewton.pdf). However I couldn't find a definition online. Does anyone know what a lower convex hull is?

Thanks in advance.

Answer 1

Informally, you take a finite set of points in $\mathbb{R}^2$, which you imagine as a set of pins. Then you tie a string to the leftmost point and wrap it tightly around the bottom of the set of pins.

Formally, consider all of the half-planes $H_u = \{v : \langle u, v \rangle \ge c\}$ with $u$ in the upper half-plane $\{u = (x, y) : y \ge 0\}$. The lower convex hull is the intersection of all these half-planes which contain your set of points. I like to call these "lower inequalities" because I picture them as lines which are below all of the points.

In the context of such Newton polygons, it is unimportant whether you are considering this as a convex set or considering just the bounded faces of that convex set.

Convex hull:

Lower convex hull: