K-弱凸性与K-弱光滑性

K-Weakly Convex and K-Weakly Smooth

提出了K_弱凸性与K_弱光滑性 ,作为K_强凸性与K_强光滑性的推广 ,然后证明了K_弱凸性与K_弱光滑性是对偶性质 ;Banach空间X是非常凸的当且仅当X是严格凸的且K_弱凸的 ;Banach空间X是局部一致凸的当且仅当X是K_强凸的和严格凸的且具有 (WM)性质。

The \%K\%_weakly convex and \%K\%_weakly smooth are defined.It is shown that \%K\%_weakly convex and \%K\%_weakly smooth are dual notions, Banach space \%X\% is very convex if and only if \%X\% is strictly convex and \%K\%_weakly convex,and Banach space \%X\% is locally uniformly convex if and only if \%X\% is \%K\%_strongly convex, strictly convex, and \%X\% has property(WM).