Abstract
Letf:V→R be a function defined on a subsetV ofR n×R d let⃜:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=⃜(x)} This paper deals with the characterization of some properties of ⃜ in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC 1 1 regularity of ⃜ in the interior ofI={x∈ R n;εR d (x t)∈V} and in theC 1 1 case an expression forD 2⃜ is given To some extent an answer is given to the following question: which convex function ⃜:I→R I interval ϒR (resp which function √:I→R of classC 1 1) is the shadow of aC 2 functionf:I×R→R?
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Ancona, A. Ombres Convexité, régularité et sous-harmonicité. Ark. Mat. 33, 1–44 (1995). https://doi.org/10.1007/BF02559604
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DOI: https://doi.org/10.1007/BF02559604