An Analysis of Zero Set and Global Error Bound Properties of a Piecewise Affine Function via Its Recession Function

For a piecewise affine function f:Rⁿ → Rᵐ, the recession function is defined by\[ f^∞ ( x ) : =limλ → ∞ f ( λ x )/λ . \] In this paper, we study the zero set and error bound properties of f via f^∞. We show, for example, that f has a zero when f^∞ has a unique zero (at the origin) with a nonvanishing index. We also characterize the global error bound property of a piecewise affine function in terms of the recession cones of the zero sets of the function and its recession function.