On the reduction of pairs of bounded closed convex sets

J. Grzybowski, D. Pallaschke, and R. Urbański. "On the reduction of pairs of bounded closed convex sets." Studia Mathematica 189.1 (2008): 1-12. <http://eudml.org/doc/284577>.

@article{J2008,
abstract = {Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ 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.},
author = {J. Grzybowski, D. Pallaschke, R. Urbański},
journal = {Studia Mathematica},
keywords = {convex analysis; separation law; pairs of bounded closed convex sets; minimal representation},
language = {eng},
number = {1},
pages = {1-12},
title = {On the reduction of pairs of bounded closed convex sets},
url = {http://eudml.org/doc/284577},
volume = {189},
year = {2008},
}

TY - JOUR
AU - J. Grzybowski
AU - D. Pallaschke
AU - R. Urbański
TI - On the reduction of pairs of bounded closed convex sets
JO - Studia Mathematica
PY - 2008
VL - 189
IS - 1
SP - 1
EP - 12
AB - Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ 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.
LA - eng
KW - convex analysis; separation law; pairs of bounded closed convex sets; minimal representation
UR - http://eudml.org/doc/284577
ER -