Can someone help me in going about proving the following?

Let S be any arbitrary set in R^n. Show that conv(S) is the smallest covex set containing S.

I was thinking a proof by contradiction. But am having a hard time getting it.

TIA :)

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- October 6th 2009, 09:47 AMginafaraOptimization theory-convex hull help
Can someone help me in going about proving the following?

Let S be any arbitrary set in R^n. Show that conv(S) is the smallest covex set containing S.

I was thinking a proof by contradiction. But am having a hard time getting it.

TIA :) - January 21st 2010, 02:44 AMRebesques
Well, if there was a convex superset D of S strictly contained in the convex hull conv(S), then there would be a convex combination of elements of S, contained in D and not contained in conv(S). This contradicts the definition of the convex hull.