
Journal of Convex Analysis 28 (2021), No. 4, 12651280 Copyright Heldermann Verlag 2021 Generous Sets Augustin Fruchard Université 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 Int. Research Center, Mathematics and Interdisciplinary Science, Shijiazhuang, P.R.China lpyuan@hebtu.edu.cn Tudor Zamfirescu Fachbereich Mathematik, Universität Dortmund, Germany, Germany and: Romanian Academy, Bucharest, Romania, and: School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China tuzamfirescu@gmail.com [Abstractpdf] \def\Rk{{\mathbb{R}^k}} We investigate the notion of generosity, a particular case of nonselfishness. Let $\cal F$ be a family of sets in $\Rk$. A set $M \subset \Rk$ 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: Fconvex, complete, generous, grateful. MSC: 52A10, 52A20. [ Fulltextpdf (821 KB)] for subscribers only. 