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In approximating the convex hull "from inside", i.e.

$ \text{conv}S = \{ x \in \mathbb{R}^n \mid x= \sum_{i=1}^k \lambda_i x^i, x^i \in S, \lambda_i \geq 0, \sum_{i=1}^k \lambda_i= 1 \} \text{,}$

in the case where $S$ is infinite - why can we restrict ourselves to finite many $x^i \in S$?

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    @hardmath I am not sure what you want to say. You can cancel out "approximation" and "inside" if this leads to confusion. It is just to seperate two ways of defining $\text{conv}S$, the one mentioned in my question, and the one $\text{conv}S = \cap_{S \subset M, M convex} M$ (this could be thought of "approximation from 'outside'"). One can define the convex hull without prior to introducing "extremal points", so I don't see why it should be relevant (Of course, for compact convex non-empty subsets of $\mathbb{R}^n$ we can reprsent the elements by a finite ($n+1$) sum of the extremal points..2012-07-30

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I assume you are really asking this: Let $S$ be an infinite subset of $\mathbb{R}^n$. Then why is the convex hull of $S$, denoted $\operatorname{conv}S$, defined as you show it, using only finite convex combinations of elements of $S$?

The answer is that the set defined this way really is convex (a convex combination of two finite convex combination is another finite convex combination), and this is the smallest convex set containing $S$ (since any convex set containing $S$ must contain any finite convex combination of members of $S$).

I guess the main point is that the definition of convexity nowhere speaks of inifinite convex combinations.

As pointed out in the comments, Carathéodory's theorem is worth mentioning: It states that you never need to take a convex combination of more than $n+1$ of the points if you work in $\mathbb{R}^n$.

There is an analogy with linear combinations: The linear span of an infinite set of vectors consists of finite linear combinations of the given vectors.

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    @Suedklee: Indeed, if you take the closed unit disk, for example, no finite number of points suffices to represent all the points in the disk via convex combinations.2012-07-30