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I was wondering what would be the largest volume possible of the shape you get if you would put 5 points on a sphere with radius r and "wrap a paper" around those points. Don't know what it's called. And I also wonder how you would position those points.

I can imagine that for 4 points they would be positioned in a way so you get a regular tetrahedron and for 8 points a cube (is this true?).

Is there a general approach for any n points?

Ivo
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    Your shape is called the "convex hull" of the points $-$ the smallest convex set that contains all the points. (Technical aside: This notion is well-defined for any bounded set of points, because the intersection of two convex sets is convex.) – TonyK Jul 24 '14 at 10:10

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This problem has been a research topic and the general answer is unknown.

The question is "who is the largest polyhedron with $n$ vertices inscribed in a unit sphere in $\mathbb R^3$?"

For $n=4$ it is the regular tetrahedron; for $n=5$ it is the union of two tetrahedra; for $n=6$ it is the octahedron, for $n=7$ it is the union of two pidamids; for $n=8$ it is NOT the cube.

Here some biblio links

http://www.ams.org/journals/mcom/1963-17-082/S0025-5718-63-99183-X/S0025-5718-63-99183-X.pdf

http://arxiv.org/pdf/1402.6496.pdf

http://www-users.cs.umn.edu/~shao/fulltext.pdf (http://link.springer.com/chapter/10.1007%2F978-3-540-44400-8_22)

user126154
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