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Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ a continuous injective function.

In general, it is not true that the image $f[X]$ is convex whenever $X \subseteq \mathbf{R}^2$ is convex.

Is there some additional assumption to ensure that $f[X]$ is convex?

  • Are you asking this to be true for all subsets $X$? I found this article about convexity preserving maps: http://arxiv.org/pdf/1212.1268.pdf – ploosu2 Mar 18 '15 at 15:27
  • Only convex sets $X$: exactly, convexity preserving maps.. –  Mar 18 '15 at 23:10

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I'm not sure if this is the level of answer that you're looking for, but there is a whole calculus of convexity-preserving operations. Stephen Boyd has some lecture slides here:

http://stanford.edu/class/ee364a/lectures/functions.pdf

In particular, affine functions are convexity preserving.

Other good sources about this would be Boyd's book "Convex Optimization" or the book "Fundamentals of Convex Analysis" by Hiriart-Urrut and Lemaréchal.

bassen
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