# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332827

Arkiv for Matematik, 33, (1), 1-44, 1993.9
Let\$f:V\$→\$R\$be a function defined on a subset\$V\$of\$R\$^{\$n\$}×\$R\$^{\$d\$}let\$⃜:x→\$inf{\$f(x t)\$;\$t\$such that\$(x t)∈V}\$denote the\$shadow\$of\$f\$and let\$Φ\$=\${(x t)∈V; f(x t)=⃜(x)}\$This paper deals with the characterization of some properties of ⃜ in terms of the infinitesimal behavior of\$f\$near points ζ∈\$Φ\$proving in particular a conjecture of J M Trépreau concerning the case\$d\$=1 Characterizations of this type are provided for the convexity the subharmonicity or the\$C\$^{1 1}regularity of ⃜ in the interior of\$I={x∈\$\$R\$^{n};ε\$R\$^{d}\$(x t)∈V}\$and in the\$C\$^{1 1}case an expression for\$D\$^{2}⃜ is given To some extent an answer is given to the following question: which convex function ⃜:\$I\$→\$R\$\$I\$interval ϒ\$R\$(resp which function √:\$I\$→\$R\$of class\$C\$^{1 1}) is the shadow of a\$C\$^{2}function\$f:I\$×\$R→R\$?
```@inproceedings{alano1993ombres,
title={Ombres Convexité, régularité et sous-harmonicité},
author={Alano Ancona},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203544834461636},
booktitle={Arkiv for Matematik},
volume={33},
number={1},
pages={1-44},
year={1993},
}
```
Alano Ancona. Ombres Convexité, régularité et sous-harmonicité. 1993. Vol. 33. In Arkiv for Matematik. pp.1-44. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203544834461636.