Geodesic
Minimal Pairs Representing Selections of Four Linear Functions in R3
Journal of convex analysis, Tome 7 (2000) no. 2, pp. 445-452.

Voir la notice de l'article provenant de la source Heldermann Verlag

We investigate minimal pairs of continuous selections of four linear functions in three dimemsional space. Our purpose is to find minimal pairs of compact convex sets (polytopes) which represent all 166 continuous selections [see S. G. Bartels, L. Kuntz and S. Scholtes, Nonlinear Analysis, Theory, Meth. Appl. 24 (1995) 385-407] which can be made from the three coordinate functions and its negative sum. We find that these 166 selections are represented by 16 essentialy different minimal pairs which were studied in previous papers of the authors [see J. Grzybowski, Arch. Math. 63 (1994) 173-181, and D. Pallaschke, S. Scholtes, R. Urbanski, Bull. Polish. Acad. Sci. Math. 39 (1991) 105-109]. Three out of 16 cases are minimal pairs that are not unique minimal representations in their own quotient classes. One of these quotient classes was already studied in the above mentioned paper of J. Grzybowski and in the papers of D. Pallaschke, R. Urbanski, Z. Oper. Res. 37 (1993) 129-150, and M. Wiernowolski, Studia Math. 124 (2) (1997) 149-154.
Classification : 52A07, 26A27
Mots-clés : Minimal pairs of convex sets, convex analysis 
@article{JCA_2000_7_2_JCA_2000_7_2_a12,
     author = {J. Grzybowski and D. Pallaschke and R. Urbanski},
     title = {Minimal {Pairs} {Representing} {Selections} of {Four} {Linear} {Functions} in {R\protect\textsuperscript{3}}},
     journal = {Journal of convex analysis},
     pages = {445--452},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2000},
     url = {http://geodesic.mathdoc.fr/item/JCA_2000_7_2_JCA_2000_7_2_a12/}
}
TY  - JOUR
AU  - J. Grzybowski
AU  - D. Pallaschke
AU  - R. Urbanski
TI  - Minimal Pairs Representing Selections of Four Linear Functions in R3
JO  - Journal of convex analysis
PY  - 2000
SP  - 445
EP  - 452
VL  - 7
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JCA_2000_7_2_JCA_2000_7_2_a12/
ID  - JCA_2000_7_2_JCA_2000_7_2_a12
ER  - 
%0 Journal Article
%A J. Grzybowski
%A D. Pallaschke
%A R. Urbanski
%T Minimal Pairs Representing Selections of Four Linear Functions in R3
%J Journal of convex analysis
%D 2000
%P 445-452
%V 7
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JCA_2000_7_2_JCA_2000_7_2_a12/
%F JCA_2000_7_2_JCA_2000_7_2_a12
J. Grzybowski; D. Pallaschke; R. Urbanski. Minimal Pairs Representing Selections of Four Linear Functions in R3. Journal of convex analysis, Tome 7 (2000) no. 2, pp. 445-452. http://geodesic.mathdoc.fr/item/JCA_2000_7_2_JCA_2000_7_2_a12/