On the reduction of pairs of bounded closed convex sets

Tom 189 / 2008

J. Grzybowski, D. Pallaschke, R. Urbański Studia Mathematica 189 (2008), 1-12 MSC: 52A07, 26A51, 49J52. DOI: 10.4064/sm189-1-1

Streszczenie

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.

Autorzy

  • J. GrzybowskiFaculty of Mathematics
    and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    PL-61-614 Poznań, Poland
    e-mail
  • D. PallaschkeInstitut für Statistik
    und Mathematische Wirtschaftstheorie
    Universität Karlsruhe
    Kaiserstr. 12
    D-76128 Karlsruhe, Germany
    e-mail
  • R. UrbańskiFaculty of Mathematics
    and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    PL-61-614 Poznań, Poland
    e-mail

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