Convex Hull 2D, 3D, anyD

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Maybe a very silly question...

I can imagine what convex hull are…in 2D and 3D (and also know –but not really understand especally 3D- some algorithms how to determine them –thx to SA– ).

Are there also convex hulls for any dimension? I assume there are, but did not found usefull information about it.

Has somebody a good link for this?

Thank you in advance.

Posted 1-Feb-14 6:58am

idle63

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CPallini · Accepted Answer · 2014-02-01T10:08:00

Solution 1

Quote:
Are there also convex hulls for any dimension?

From Convex Hull Wikipedia[^]a page:

"Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces; they may also be generalized further, to oriented matroids."

That's a clue convex hulls exist for arbitrary dimensions (please note, that does not mean I understand it :-) ).

In such Wikipedia page you may also find useful references.

Posted 1-Feb-14 10:08am

CPallini

Comments

[no name] 1-Feb-14 16:23pm

Why I do not find this with google...Thank you very much! Please note I also do not understand all this stuff, but I have to implement it :( :)

CPallini 1-Feb-14 16:33pm

:-) I know that feeling. Good luck!

Sergey Alexandrovich Kryukov 1-Feb-14 22:16pm

That is quite correct and even seems to be pretty obvious, my 5.
However, the algorithmic solution for finding a hull is much less trivial thing. I would speculate that the existing solution (such as AForge.NET 2D solution I advised OP last time) could be naturally generalized to higher dimensions, but I never looked at such problem so far...
—SA

CPallini 2-Feb-14 3:24am

Of course. Knowing it exists is just a starting point and the problem actually looks pretty complex.
By the way, thank you.

Sergey Alexandrovich Kryukov 2-Feb-14 12:11pm

Agree; how much really complex — not so sure; I would need time to dig into it. There are many cases where generalization for nD is just trivial, maybe in most cases, but also there is the famous Poincaré conjecture, one of the Millenium Prize Problems, proven by Perelman only in 2002, with great difficulty. Analogous statements for manifolds less then 3D was proven a long time ago, generalized for more then 3D, and only 3D was a hard die...
—SA