TY - JOUR
T1 - Regularity and integration of set-valued maps represented by generalized Steiner points
AU - Baier, Robert
AU - Farkhi, Elza
PY - 2007/6
Y1 - 2007/6
N2 - A family of probability measures on the unit ball in ℝn generates a family of generalized Steiner (GS-)points for every convex compact set in ℝn. Such a "rich" family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.
AB - A family of probability measures on the unit ball in ℝn generates a family of generalized Steiner (GS-)points for every convex compact set in ℝn. Such a "rich" family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.
KW - Arithmetic set operations
KW - Aumann integral
KW - Castaing representation
KW - Demyanov distance
KW - Generalized Steiner selections
KW - Set-valued maps
UR - http://www.scopus.com/inward/record.url?scp=34247850699&partnerID=8YFLogxK
U2 - 10.1007/s11228-006-0038-0
DO - 10.1007/s11228-006-0038-0
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:34247850699
SN - 1877-0533
VL - 15
SP - 185
EP - 207
JO - Set-Valued and Variational Analysis
JF - Set-Valued and Variational Analysis
IS - 2
ER -