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Abstract and Applied Analysis
Volume 2005, Issue 5, Pages 469-488

Small sets in convex geometry and formal independence over ZFC

Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

Received 28 June 2004

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


To each closed subset S of a finite-dimensional Euclidean space corresponds a σ-ideal of sets 𝒥(S) which is σ-generated over S by the convex subsets of S. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals and other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self-maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. We also attempt to present to nonspecialists the set-theoretic methods for dealing with formal independence as a means of geometric investigations.