Geometry of Orlicz spaces equipped with norms generated by some lattice norms in ℝ^2

In Orlicz spaces generated by convex Orlicz functions a family of norms generated by some lattice norms in ℝ^2 are defined and studied. This family of norms includes the family of the p-Amemiya norms ( 1≤ p≤∞ ) studied in Cui et al. (Nonlinear Anal. 69:1796–1816, 2008 ; Nonlinear Anal. 71:6343–6364, 2009 ; J. Math. Anal. Appl. 432:1095–1105, 2015 ; Nonlinear Anal. 75:3973–3993, 2012 ) and He et al. (Fixed Point Theory Appl. 2013:1–18, 2013 ). Criteria for strict monotonicity, lower and upper local uniform monotonicities and uniform monotonicities of Orlicz spaces and their subspaces of order continuous elements, equipped with these norms, are given in terms of the generating Orlicz functions, and the lattice norms in ℝ^2 . The problems of strict convexity and of the existence of order almost isometric as well as order isometric copies in these spaces are also discussed.