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Thread: closed and convex sets

  1. #1
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    closed and convex sets

    How does one proof the closeness in this case?

    the questions is show that $\displaystyle (C)_{\mathbf{v}}$ is closed and convex if $\displaystyle C$ is.

    $\displaystyle (C)_{\mathbf{v}}$ = {x : x = $\displaystyle \mathbf{v}$ + y for some y in $\displaystyle C$ }

    To prove that $\displaystyle (C)_{\mathbf{v}}$ is convex i did this:

    Let $\displaystyle x_{1}, x_{2}$ be in $\displaystyle (C)_{\mathbf{v}}$, then

    $\displaystyle x_{1} = \mathbf{v} + y_{1}$ and
    $\displaystyle x_{2} = \mathbf{v} + y_{2}$ , $\displaystyle y_{1}, y_{2}$ in $\displaystyle C$.

    thus,
    $\displaystyle x = \mu x_{1} + (1 - \mu)x_{2}$
    $\displaystyle x =\mathbf{v} + \mu y_{1} + (1 - \mu)y_{2}$

    but we know that $\displaystyle C$ is convex so
    $\displaystyle \mu y_{1} + (1 - \mu)y_{2}$ is in $\displaystyle C$

    thus
    $\displaystyle x =\mathbf{v} + \mu y_{1} + (1 - \mu)y_{2}$ is in $\displaystyle (C)_{\mathbf{v}}$.

    but how does one prove that $\displaystyle C_{\mathbf{v}$ is closed?
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  2. #2
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    Quote Originally Posted by mtmath View Post
    How does one proof the closeness in this case?

    the questions is show that $\displaystyle (C)_{\mathbf{v}}$ is closed and convex if $\displaystyle C$ is.

    $\displaystyle (C)_{\mathbf{v}}$ = {x : x = $\displaystyle \mathbf{v}$ + y for some y in $\displaystyle C$ }

    To prove that $\displaystyle (C)_{\mathbf{v}}$ is convex i did this:

    Let $\displaystyle x_{1}, x_{2}$ be in $\displaystyle (C)_{\mathbf{v}}$, then

    $\displaystyle x_{1} = \mathbf{v} + y_{1}$ and
    $\displaystyle x_{2} = \mathbf{v} + y_{2}$ , $\displaystyle y_{1}, y_{2}$ in $\displaystyle C$.

    thus,
    $\displaystyle x = \mu x_{1} + (1 - \mu)x_{2}$
    $\displaystyle x =\mathbf{v} + \mu y_{1} + (1 - \mu)y_{2}$

    but we know that $\displaystyle C$ is convex so
    $\displaystyle \mu y_{1} + (1 - \mu)y_{2}$ is in $\displaystyle C$

    thus
    $\displaystyle x =\mathbf{v} + \mu y_{1} + (1 - \mu)y_{2}$ is in $\displaystyle (C)_{\mathbf{v}}$.

    but how does one prove that $\displaystyle C_{\mathbf{v}$ is closed?


    It'd be great if you'd tell us what is C: is this a vector/metric/norm/in general, topological space or what?

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    It'd be great if you'd tell us what is C: is this a vector/metric/norm/in general, topological space or what?

    Tonio
    Actually the whole question reads as:

    Let $\displaystyle (C)_{\mathbf{v}}$ denote the translation of a set $\displaystyle C$ by vector $\displaystyle \mathbf{v}$, i.e.,

    $\displaystyle (C)_{\mathbf{v}} \triangleq \{x: x = \mathbf{v}$ + y for some y $\displaystyle \in C\}.$

    Show that $\displaystyle (C)_{\mathbf{v}}$ is also closed and convex if $\displaystyle C$ is.
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  4. #4
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    Quote Originally Posted by mtmath View Post
    Actually the whole question reads as:

    Let $\displaystyle (C)_{\mathbf{v}}$ denote the translation of a set $\displaystyle C$ by vector $\displaystyle \mathbf{v}$, i.e.,

    $\displaystyle (C)_{\mathbf{v}} \triangleq \{x: x = \mathbf{v}$ + y for some y $\displaystyle \in C\}.$

    Show that $\displaystyle (C)_{\mathbf{v}}$ is also closed and convex if $\displaystyle C$ is.


    I understood the question from the beginning: where are C and v taken from?! What's the question's context?

    Tonio
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  5. #5
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    Quote Originally Posted by tonio View Post
    I understood the question from the beginning: where are C and v taken from?! What's the question's context?

    Tonio
    I meant to say this is an exercise after a chapter on projections Onto convex sets. $\displaystyle C$ and $\displaystyle \mathbf{v}$ are not defined on the question.
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  6. #6
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    Quote Originally Posted by mtmath View Post
    I meant to say this is an exercise after a chapter on projections Onto convex sets. $\displaystyle C$ and $\displaystyle \mathbf{v}$ are not defined on the question.

    C must be defined as some kind of topological structure, otherwise I can't see how "closedness" can

    be defined...

    Tonio
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  7. #7
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    Quote Originally Posted by tonio View Post
    C must be defined as some kind of topological structure, otherwise I can't see how "closedness" can

    be defined...

    Tonio
    $\displaystyle C$ is not defined anywhere in the question. So should i take my answer as a complete answer to the question?
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