Convex hull of the Mandelbrot set

Question

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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I don't think this is known, neither the convex hull nor even the points with largest real part, i.e., the rightmost points. — Lukas Geyer, Dec 19 '14 at 07:01
@LukasGeyer, do you know a reference where this is discussed? — lhf, Dec 19 '14 at 11:20

score 4 · Accepted Answer · edited Apr 13 '17 at 12:21

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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.

edited Apr 13 '17 at 12:21

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answered Dec 19 '14 at 14:06

Sheldon L

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Oh, so $\pm i$ are not extreme points! – lhf Dec 19 '14 at 15:16
yeah, I added the approximate coordinates of the imaginary extreme in the answer. – Sheldon L Dec 19 '14 at 15:27
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@lhf, given the self-similar fractal nature of these Mandelbrot extremum points, I wonder if there might be an infinite number of points of contact with the convex hull... – Sheldon L Dec 19 '14 at 18:24

Convex hull of the Mandelbrot set

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