A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations

\def\R{\mathbb R} Let $\R^n$ denote the usual n-dimensional Euclidean space. A polyhedral convex function $f \colon \R^n \to \R\cup\{+\infty\}$ can always be seen as the pointwise limit of a certain family $\{f^t\}_{t>0}$ of $C^{\infty}$ convex functions. An explicit construction of this family $\{f^t\}_{t>0}$ can be found in a previous paper by the second author [A. Seeger, Smoothing a polyhedral convex function via cumulant transformation and homogenization, Annales Polinici Mathematici 67 (1997) 259--268]. The aim of the present work is to further explore this $C^{\infty}$-approximation scheme. In particular, one shows how the family $\{f^t\}_{t>0}$ yields first and second-order information on the behavior of $f$. Links to linear programming and Legendre-Fenchel duality theory are also discussed.