Normal Maps Induced by Linear Transformations

We study a certain piecewise linear manifold, which we call the normal manifold, associated with a polyhedral convex set, and a family of continuous functions, called normal maps, that are induced on this manifold by continuous functions from Rⁿ to Rⁿ. These normal maps occur frequently in optimization and equilibrium problems, and the subclass of normal maps induced by linear transformations plays a key role.

Our main result is that the normal map induced by a linear transformation is a Lipschitzian homeomorphism if and only if the determinant of the map in each n-cell of the normal manifold has the same (nonzero) sign.