# Mappings preserving convex polyhedra

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