On the reduction of pairs of bounded closed convex sets
Volume 189 / 2008
Studia Mathematica 189 (2008), 1-12
MSC: 52A07, 26A51, 49J52.
DOI: 10.4064/sm189-1-1
Abstract
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.
