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How do you show that the convex hull of a given set of points S, always has the minimum perimeter ? By perimeter i mean the length of the boundary of the hull

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    Minimum with respect to what? To all convex sets containing S? Cause if its w.r.t. all sets containing S, then it is not true. Take for instance the 4 corners of a square. obviously, the square is the convex hull, its boundary has length $4Z$, $Z$ being the length of one side. But the set of corner points itself has zero length.2010-12-05
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    I mean that the hull has the shortest perimeter of all simple polygons that include every point of S.2010-12-05

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