Abstract: In this paper, via Bochner integral of vector-valued functions, the authors introduce the concepts of integral convex sets and integral convex functionals and integral extremal points of sets in Banach spaces. The authors mainly show that every finite dimensional convex set and every open or closed convex set are integral convex; every lower semi-continuous convex functional and every upper semi-continuous convex functional defined on a open convex set are integral convex; every nonempty compact sets have integral extremal points; the integral extremal points set is equal to the extremal points set for every compact convex set. Two applications of integral convexity are obtained at last.

Key words: Bochner integral, Integral convex set, Integral convex functional, Integral extremal point, Integral convex programming