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Math Help - Convex bodies Shadow.

  1. #1
    Member kezman's Avatar
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    Convex bodies Shadow.

    For  X  \subseteq   E^3
    and u \in E^3

    we set
    S(u;X) = \left \{ tu + x: t > 0,x \in X \right \}
    S(u;X) is the shadow of X when the light is coming from the direction u.
    Let u \in E^3

    .Prove .

    (a) If X is a convex subset of E^3
    , then S(u;X) is also convex.

    (b) If X is open in E^3

    , then S(u;X) is open and  X \subseteq  S(u;X).



    Im not sure how to aproach both of them.

    Thanks.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Convex bodies Shadow.

    Well for
    For
    a) Any element of S(u;X) has the form  ut + x \text{ where } t >0 \text{ and } x \in X Well, to show convexity you have to show that for any  p \in [0,1] p(tu + x) + (1-p)(tu + x) \in S(u;X) So multiplying the p through we get  ptu + px + (1-p)tu + (1-p)x since X is convex,  px + (1-p)x = j \in X . So we got  ptu + (1-p)tu + j , then factor out the t to get  (pu+(1-p)u)t + j , which is  (pu + u - pu)t + j which is  ut + j \in S(u; X) since  j \in X

    b)
    Assume X is open.
    To show that S(u;X) is open, for any element  j \in S(u;X) i need to show that there exists a positive radius r such that this implication holds, if p \in B_r(j) then p must be in S(u;X).
    Now, since  j \in S(u;X) j is of the form  j = tu + x where  x \in X . Since X is open, there exists a radius  r_1 > 0 such that this implication holds, if  \bar{x} \in B_{r_1}(x) then  \bar{x} \in X .

    Now i will show that for the radius  r_1 > 0 derived from j = tu + x (it is the radius for x, an element of X ), if  p \in B_{r_1}(j) then  p \in S(u;X) .
    Now, assume  p \in B_{r_1}(j) So  ||p -(ut + x) || = ||p - ut - x || < r_1 . So this means that  ||(p-ut) - x || < r_1 which means  p - ut \in X Thus  ut + (p-ut) \in S(u;X) , thus  p \in S(u; X) which means that S(u;X) is open.
    Last edited by jakncoke; December 29th 2012 at 04:57 PM.
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  3. #3
    Member kezman's Avatar
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    Re: Convex bodies Shadow.

    thanks for the help jakncoke !!!!!!
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  4. #4
    Member kezman's Avatar
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    Re: Convex bodies Shadow.

    I re open the thread, cause im missing the inclusion X in S(u,X) knowing that X is open. I have no clue.
    Thanks!
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