This paper is a continuation of the paper [Y. Cui, L. Duan, H. Hudzik, M. Wisła, Basic theory of p-Amemiya norm in Orlicz spaces (1≤p≤∞): Extreme points and rotundity in Orlicz spaces equipped with these norms, Nonlinear Anal. 69 (2008) 1796–1816] and an extension of the paper [Y. Cui, T. Wang, Strongly extreme points of Orlicz space, J. Math. 4 (1987) 335-340]. Criteria for the Kadec–Klee property with respect to the convergence in measure and for strongly extreme points in Orlicz spaces LΦ,p equipped with thep-Amemiya norm (1≤p≤∞) are given. Orlicz spaces with nonempty (and empty) set of strongly extreme points of the unit ball B(LΦ,p) are characterized. It is interesting to note that for some Orlicz functions Φ there is a critical number p0 such that the problem of emptiness or nonemptiness of strongly extreme points of the unit ball B(LΦ,p) looks different for 1≤p≤p0 and for p0<p≤∞. Criteria for the midpoint local uniform rotundity of the spaces LΦ,p are also deduced. Theorems stated in this paper, even restricted to the case p∈{1,∞}, are more general than the results of the latter reference mentioned above because the assumption limu→∞Φ(u)/u=∞ was not used. The methods used in the proofs are universal for all 1≤p≤∞. New notions of p-spherical convergence and H2,μ-points are introduced and considered. Moreover, Orlicz spaces LΦ,p which are linearly isomorphic or isometric to L∞ are characterized as auxiliary techniques and facts for some proofs.