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Please give me an example to show this. Thank you!

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closed as off-topic by Antonio Vargas, drhab, M Turgeon, Daniel Fischer, Rick Decker Aug 3 at 18:25

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Please provide more context to your question: What is the definition of "convexity closure"? Why do you ask the question, i.e. what is the problem that you are thinking about, or the aspect that you would like to understand? –  Tim van Beek Jun 11 '11 at 9:24
Convex hull, you mean? –  Grigory M Jun 11 '11 at 10:19

1 Answer 1

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Ok, I have seen your other question about the meaning of convexity, so I guess you were asking about that. Since we are talking about convexity, we need a vector space. Since we are talking about closure, we actually need a topological vector space, which is a vector space that is also a topological space, such that the algebraic operations are continuous. A very important concept in the theory is the concept of convex sets $C$, which is a set such that when $x ,y \in C$, then $(\lambda x + (1- \lambda) y) \in C$ for all $0 \le \lambda \le 1$.

A lot of the theory of topological vector spaces is about locally convex spaces, which are spaces such that every point $x$ has an open neighborhood $C$ that is convex.

Another important concept is "convex hull of a given set", which for a given set $B$ is the smallest set $C \supset B$ that is convex. I guess that the "convex closure" of your question refers to the closure of the convex hull of a given set.

When you are handed a subset $B$ of a topological vector space $V$ and need to form the "closure of the convex hull", you will first have to form convex sums of all the elements, that is $$ y := \sum_{i = 0}^{n} \lambda_i x_i $$ with elements $x_i \in V$, and scalars $\lambda_i$ such that $$ \sum_{i = 0}^{n} \lambda_i = 1 $$ All of these elements will form the convex hull of $B$. Then you'll need to determine the closure of this set. This depends on the topology of $V$.

A simple example is of course $B = \{0, 1\}$ as a subset of the topological vector space $\mathbb{R}$. Then the convex hull is $[0, 1]$. This is already closed in the canonical topology of $\mathbb{R}$.

If you would like to see an example that is more sophisticated, please formulate a question that provides more context :-)

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Thank you for detailed answer. Could you show me that $\lambda_{1}x+\lambda_{1}y$ is equivalent to $\lambda_{1}x_{1}+...+\lambda_{n}x_{n}$ for convexity? Because I have not understood this. –  zghu001 Jun 11 '11 at 11:24
@zghu001: Sorry, I don't quite understand your question. The definition of a convex set involves the sum of two elements. When you iterate this, i.e. add a third, then a fourth etc. you end up with the convex sum involving n elements, that I wrote down in my answer. –  Tim van Beek Jun 12 '11 at 6:24

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