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singhar
Newbie

Posts: 22
 Convex sets vs. Affine sets   « on: Mar 29th, 2010, 5:12pm » Quote Modify

Hi,

I have a very fundamental doubt. Would really appreciate if anyone could help me understand why every affine set is also a convex set?

Per defintion, in both affine and convex sets the coefficients in the linear combination of points (which must also belong to the convex or affine set) must add to 1, but in case of convex set there is the extra condition that the coefficients must be positive. So doesn't that make the definition of convex sets more restricted than affine sets?

And thus how can every affine set be a convex set? Should it not be other way round?
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Obob
Senior Riddler

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Posts: 489
 Re: Convex sets vs. Affine sets   « Reply #1 on: Mar 29th, 2010, 7:52pm » Quote Modify

A set X is affine if given x1,...,xn in X and real numbers a1,...,an with a1+...+an = 1, then a1*x1+...+an*xn is in X.

A set X is convex if given x1,...,xn in X and positive real numbers a1,...,an with a1+...+an = 1, then a1*x1+...an*xn is in X.

If X is affine, then if a1,...,an are positive real numbers with a1+...+an = 1 then they are also arbitrary numbers summing to 1, so if x1,...,xn are in X then a1*x1+...+an*xn is in X.  Thus X is convex.

Geometrically, affine sets are always lines, planes, 3-planes, etc.  They are clearly convex.
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singhar
Newbie

Posts: 22
 Re: Convex sets vs. Affine sets   « Reply #2 on: Mar 30th, 2010, 10:27am » Quote Modify

Thanks, Obob, I kind of realized where my thoughts were getting messed up.
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