中文摘要：

本文证明了k极凸是严格介于冼军和胡长松的k极凸性和何仁义的k极凸性之间的一种新凸性.利用k极凸空间的概念,得到了k极凸的性质以及与其它凸性之间的蕴涵关系,完善了k极光滑及其对偶空间的研究.

英文摘要：

In this article, we show that the k-extrem convexity introduced by us is just the notion of duality of Xian Jun and Hu Changsong’s k-extrem smoothness and lies strictly between the Xian Jun and Hu Changsong’s notion of k-extrem convexity and He Renyi’s notion of k-extrem convexity. Using the notion of k-extrem convexity, we obtain some properties of this kind of convexity and study the relation between that and various convexities. The results perfect the research on k-extrem smooth space and its dual space.