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Journal of Convex Analysis 28 (2021), No. 4, [final page numbers not yet available]
Copyright Heldermann Verlag 2021



Generous Sets

Augustin Fruchard
Universitť de Haute Alsace, IRIMAS, Mulhouse, France
augustin.fruchard@uha.fr

Liping Yuan
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China
and: Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, P.R.China
lpyuan@hebtu.edu.cn

Tudor Zamfirescu
Fachbereich Mathematik, Universitšt Dortmund, Germany, Germany
and: Roumanian Academy, Bucharest, Roumania, and: School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China
tuzamfirescu@gmail.com



[Abstract-pdf]

\def\R{{\mathbb{R}}} We investigate the notion of generosity, a particular case of non-selfishness. Let $\cal F$ be a family of sets in $\R$. A set $M \subset \R$ is called $\cal F$-{\it convex} if for any points $x,y\in M$ there is a set $F\in \cal F$ such that $x,y\in F$ and $F\subset M$. We call a family $\cal F$ of compact sets {\it complete} if $\cal F$ contains all compact $\cal F$-convex sets. A single convex body $K$ will be called {\it generous}, if the family of all convex bodies isometric to $K$ is not complete. We investigate here the generosity of convex bodies.

Keywords: F-convex, complete, generous, grateful.

MSC: 52A10, 52A20.

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