Higher-order smoothing technique for polyhedral convex functions: geometric and probabilistic considerations

This Note deals with a higher order smoothing technique for a polyhedral co nvex function f : R-n --> R boolean OR{+infinity}. This technique consists in approximating f by a family {f(t)}(t>0), with f(t) :R-n --> R being conv ex and infinitely often differentiable. The explicit formula for f(t) is gi ven in terms of a function M : R-n x R --> R whose expression is derived st raightforwardly from the canonical representation of f. We show that M gene rates a wealth of information on rite behaviour of f. (C) 2000 Academie des sciences/Editions scientifiques ct medicales Elsevier SAS.