On the reduction of pairs of bounded closed convex sets

J. Grzybowski ¹   ; D. Pallaschke ²   ; R. Urbański ¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 PL-61-614 Poznań, Poland
² Institut für Statistik und Mathematische Wirtschaftstheorie Universität Karlsruhe Kaiserstr. 12 D-76128 Karlsruhe, Germany

Studia Mathematica, Tome 189 (2008) no. 1, pp. 1-12

Résumé

Let $X$ be a Hausdorff topological vector space. For nonempty bounded closed convex sets $A,B,C,D \subset X$ we denote by $A \mathbin {\dotplus }B$ the closure of the algebraic sum $A + B$, and call the pairs $(A,B)$ and $(C,D)$ equivalent if $A \mathbin {\dotplus }D = B \mathbin {\dotplus }C$. We prove two main theorems on reduction of equivalent pairs. The first theorem implies that, in a finite-dimensional space, a pair of nonempty compact convex sets with a piecewise smooth boundary and parallel tangent spaces at some boundary points is not minimal. The second theorem generalizes and unifies two main techniques of reduction of pairs of compact convex sets.

Détail

DOI : 10.4064/sm189-1-1

Keywords: hausdorff topological vector space nonempty bounded closed convex sets d subset denote mathbin dotplus closure algebraic sum call pairs equivalent mathbin dotplus mathbin dotplus prove main theorems reduction equivalent pairs first theorem implies finite dimensional space pair nonempty compact convex sets piecewise smooth boundary parallel tangent spaces boundary points minimal second theorem generalizes unifies main techniques reduction pairs compact convex sets