Ombres. Convexité, régularité et sous-harmonicité

Abstract

Let $f : V → \mathbf{R}$ be a function defined on a subset $V$ of $\mathbf{R}^n × \mathbf{R}^d$, let $\varphi : x \mapsto \inf \lbrace f(x, t); t$ such that $(x, t) ∈ V \rbrace$ denote the shadow of $f$, and let $Φ = \lbrace (x, t) ∈ V; f(x, t) = \varphi(x) \rbrace$. This paper deals with the characterization of some properties of $\varphi$ in terms of the infinitesimal behavior of $f$ near points $\xi ∈ Φ$, 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 $\varphi$ in the interior of $I = \lbrace x ∈ \mathbf{R}^n; \exists t \in \mathbf{R}^d, (x,t) ∈ V \rbrace$, and, in the $C^{1,1}$ case, an expression for $D^2 \varphi$ is given. To some extent, an answer is given to the following question: which convex function $\varphi : I → \mathbf{R}, I$ interval $\subset \mathbf{R}$ (resp. which function $\varphi : I → \mathbf{R}$ of class $C^{1,1}$) is the shadow of a $C^2$ function $f:I × \mathbf{R}→\mathbf{R}$?