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It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra.

Can you give a characterization of the class of mappings that preserve convex polyhedra?

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I think at the end you'll find that the only such mappings are affine linear. To see that the mappings must at least be locally linear, observe that any small piece of hypersurface must be mapped to a small piece of hypersurface. To show that the mapping may not be piecewise linear, you'll need to cook up some convex polyhedron that loses convexity. –  A Blumenthal Mar 6 '13 at 17:58
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Note that any continuous function from $\mathbb{R}$ to $\mathbb{R}$ preserves convex polyhedra and don't need to be affine. But I guess in higher dimension, if the image of the map is not contain in a line then your map is affine. So far I did not find clear arguments to prove it. I don't think this is easy. –  Gilles Bonnet May 5 at 19:43
    
@ABlumenthal, any idea to show rigurosly that small piece of hypersurface are send to small pieces of hypersurface? How to show that the map is continuous? –  Gilles Bonnet May 5 at 19:45

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