Geodesic
A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations
Journal of convex analysis, Tome 8 (2001) no. 1, pp. 109-126.

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\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.
Classification : 41A30, 52B70, 60E10
Mots-clés : Polyhedral convex function, smooth approximation, subgradient, linear programming 
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     author = {S. Guillaume and A. Seeger},
     title = {A {Higher-Order} {Smoothing} {Technique} for {Polyhedral} {Convex} {Functions:} {Geometric} and {Probabilistic} {Considerations}},
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S. Guillaume; A. Seeger. A Higher-Order Smoothing Technique for Polyhedral Convex Functions: Geometric and Probabilistic Considerations. Journal of convex analysis, Tome 8 (2001) no. 1, pp. 109-126. http://geodesic.mathdoc.fr/item/JCA_2001_8_1_JCA_2001_8_1_a4/