A note on integral-convexity in Banach spaces

In the present paper we focus on a generalization of the notion of integral convexity. This concept, introduced in [J.Y.Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J.Math. Anal. Appl. 295 (2004) 211–224] by replacing, in the definition of classical notion of convexity, the sum by the integral, has interesting applications in optimal control problems. By using, instead of Bochner integral, a more general vector integral, that of Pettis, we obtain some results on integral-extreme points of subsets of a Banach space stronger than those given in [J.Y. Wang, Y.M. Ma, The integral convexity of sets and functionals in Banach spaces, J. Math. Anal. Appl. 295 (2004) 211–224]. Finally, a natural example coming from measure theory is included, in order to reflect the relationships between different kinds of integral convexity. © 2006 Elsevier Inc. All rights reserved