Compositions, Generated by Special Nets in Affinely Connected Spaces

Special nets which characterize Cartesian, geodesic, Chebyshevian, geodesic- Chebyshevian and Chebyshevian-geodesic compositions are introduced. Con- ditions for the coefficients of the connectedness in the parameters of these special nets are found.

Conjugate Compositions in Even-Dimensional Affinely Connected Spaces without a Torsion

Let in even-dimensional a±nely connected space without a torsion A2m be given a composition Xm£Xm by the affinor a¯ ®. The affinor b¯ ®, determined with the help of the eigen-vectors of the matrix (a¯ ®), de¯nes the second composition Ym £ Y m. Conjugate compositions are introduced by the condition: the a±nors of any of both compositions transform the vectors from the one position of the composition, generated by the other a±nor, in the vectors from the another its position. It is proved that the compositions de¯ne by a±nors a¯ ® and b¯ ® are conjugate. It is proved also that if the composition Xm£Xm is Cartesian and composition Ym£Y m is Cartesian or chebyshevian, or geodesic than the space A2m is affine.

Special compositions in affinely connected spaces without a torsion

2000 Mathematics Subject Classification: 53B05, 53B99.

Co-Connected Spaces

∗ Financial support of the Grant Agency of the Czech Republic under the grant no 201/96/0119 and of the Grant Agency of the Charles University under the grant GAUK 149 is gratefully acknowledged.

Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

* This work was supported by the CNR while the author was visiting the University of Milan.