Convex hull of the Mandelbrot set

What is the convex hull of the Mandelbrot set?

I know that the leftmost point is $c=-2$ and I thought the extreme vertical points were $c=\pm i$. Sheldon's answers says they're not.

I think that the line segments defined by $-2$ and the extreme vertical points are edges of the convex hull, but the edges on the right are harder. In particular, I don't know the extreme horizontal points on the right.

I wonder whether the convex hull of the Mandelbrot set is a convex polygon, that is, has a finite number of vertices.

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Solution 1:

There is a previous mathstack question on the largest imaginary point of the Mandelbrot set, which is $z \approx -0.207107867093967+1.122757063632597i$

Supremum of all y-coordinates of the Mandelbrot set

Also see mrob's website, for both the northernmost point and the easternmost point http://mrob.com/pub/muency/northernmostpoint.html http://mrob.com/pub/muency/easternmostpoint.html

There are several other pertinent convex hull points on the right of the Mandelbrot set, but I wouldn't know how to generate them.