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Shadows. Convexity, regularity, and subharmonicity. (Ombres. Convexité, régularité et sous-harmonicité.) (French) Zbl 0842.31004

The following situation is considered in the paper: let \(W\) be an open set, \(W\in \mathbb{R}^n \times \mathbb{R}^d\), \((n,d\geq 1)\) and \(f: W\to \mathbb{R}\) a \(C^{d+1}\) class function on \(W\). The function \(\varphi (x)= \inf \{ f(x, t)\), \(t\in \mathbb{R}^d\), \((x, t)\in W\}\) is called the shadow of \(f(x, t)\), the set \(\varphi= \{(x, t)\in W\); \(\varphi (x)= f(x, t)\}\) is called the contact set of \(f(x, t)\). The set \(\widetilde {\varphi} \subset \varphi\) is a full set of \(W\) if for every \(x\in \mathbb{R}^d\), there exists \(t\in \mathbb{R}^d\) so that \((x, t)\in \widetilde {\varphi}\).
Generalizing some results of J. M. Trépreau contained in his thesis (Reims 1984) and in an unpublished paper (1993), the author searches for conditions under which the shadow of \(f(x, t)\) is a subharmonic function. He analyzes all the possibilities respecting \(n\) and \(d\): \((n=1\), \(d=1)\), \((n>1\), \(d=1)\), \((n=1\), \(d\geq 2)\), \((n\geq 1\), \(d\geq 1)\), giving also many other results; convexity of \(\varphi (x)\), its regularity, the first and second derivatives of the shadow. The paper is very rich in results, comments of the results, comparisons between the author’s results and other ones obtained previously by other authors.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
Full Text: DOI

References:

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