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Math Help - Convex compact sets

  1. #1
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    Convex compact sets

    Given a Euclidean n-space, take any compact convex set S in that space, with a distinguished point p in its interior. Let \lambda be any positive real, and let S^{\prime} be the result of scaling S as centered on p by the factor \lambda. I would have thought that S^{\prime} must be compact and convex as well. However, I didn't manage prove this. Is there any easy proof for this?
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  2. #2
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    Quote Originally Posted by Richard View Post
    Given a Euclidean n-space, take any compact convex set S in that space, with a distinguished point p in its interior. Let \lambda be any positive real, and let S^{\prime} be the result of scaling S as centered on p by the factor \lambda. I would have thought that S^{\prime} must be compact and convex as well. However, I didn't manage prove this. Is there any easy proof for this?
    I don't think that it is necessary for P to be an interior point of S to prove it.
    First, compactness: if S' has an open cover, then the inversely scaled cover is an open cover of S, hence has a finite subcover for S, and the scaled version of this is a finite subcover of the original open cover of S'.
    Second, convexity: the inverse scaling of two points of S', x'_1,x'_2 say, are two points x_1,x_2 of S. If S is convex, then the entire line segment between these inversely scaled points must be in S, and therefore the scaled version of that line segment must be in S'. So to really show this in detail, you need to show that the scaled segment between x_1 and x_2 is identical to the line segment between x'_1 and x'_2.
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