Journal of Convex Analysis 28 (2021), No. 4, 1265--1280
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
[Abstract-pdf]
\def\Rk{{\mathbb{R}^k}} We investigate the notion of generosity, a particular case of non-selfishness. 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: F-convex, complete, generous, grateful.
MSC: 52A10, 52A20.
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