Abstract

We investigate the notion of generosity, a particular case of non-selfishness. Let F\cal F be a family of sets in Rk\mathbb{R}^k. A set M⊂RkM \subset \mathbb{R}^k is called F\cal F-convex if for any points x,y∈Mx,y\in M there is a set F∈FF\in \cal F such that x,y∈Fx,y\in F and F⊂MF\subset M. We call a family F\cal F of compact sets complete if F\cal F contains all compact F\cal F-convex sets. A single convex body KK will be called generous, if the family of all convex bodies isometric to KK is not complete. We investigate here the generosity of convex bodies.

Author information

Contact details are reproduced from the original publication and may be historical.

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
and: Romanian Academy, Bucharest, Romania,
and: School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China

tuzamfirescu@gmail.com

Keywords

  • F-convex
  • complete
  • generous
  • grateful

Mathematics Subject Classification

52A10, 52A20

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Suggested citation

A. Fruchard, L. Yuan, T. Zamfirescu. “Generous Sets.” Journal of Convex Analysis 28 (2021), No. 4, 1265–1280. https://doi.org/10.68381/jca28074

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press.