A higher-order smoothing technique for polyhedral convex functions: Geometric and probabilistic considerations

Let R-n denote the usual n-dimensional Euclidean space. A polyhedral convex function f : R-n --> R boolean OR{+infinity} can always be seen as the poi ntwise limit of a certain family {f(t)}(t>0) of C-infinity convex functions . An explicit construction of this family {f(t)}(t>0) can be found in a pre vious paper by the second author. The aim of the present work is to further explore this C-infinity-approximation scheme. In particular, one shows flo w the family {f}(t>0) yields first and second-order information on the beha vior of f. Links to linear programming and Legendre-Fenchel duality theory are also discussed.