DOI: 10.68381/jca08005 ISSN: 0944-6532
A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations
Sophie Guillaume, Alberto Seeger
Let
\mathbb{R}^n
R
n
denote the usual n-dimensional Euclidean space. A polyhedral convex function
f \colon \mathbb{R}^n \to \mathbb{R}\cup\{+\infty\}
f
:
R
n
→
R
∪
{
+
∞
}
can always be seen as the pointwise limit of a certain family
\{f^t\}_{t>0}
{
f
t
}
t
>
0
of
C^{\infty}
C
∞
convex functions. An explicit construction of this family
\{f^t\}_{t>0}
{
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}
C
∞
-approximation scheme. In particular, one shows how the family
\{f^t\}_{t>0}
{
f
t
}
t
>
0
yields first and second-order information on the behavior of
f
f
. Links to linear programming and Legendre-Fenchel duality theory are also discussed.