In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898-907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using $\psi$-direct sums of two Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on $[0,1]$. In this paper, we characterize the $B_n$-convexity and $J_n$-convexity of Banach spaces using $\psi$-direct sums of $n$ Banach spaces, where $\psi$ is a continuous convex function with some appropriate conditions on a certain convex subset of $\mathbb R^n$.
References
A. Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. MR133857 10.2307/2034494 0108.31401A. Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. MR133857 10.2307/2034494 0108.31401
D.H. Hyers, G. Isac and Th.M. Rassias, Topics in nonlinear analysis and applications. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR1453115D.H. Hyers, G. Isac and Th.M. Rassias, Topics in nonlinear analysis and applications. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR1453115
R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550. MR173932 10.2307/1970663 0132.08902R. C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550. MR173932 10.2307/1970663 0132.08902
M. Kato, K.-S. Saito and T. Tamura, On $\psi$-direct sums of Banach spaces and convexity, J. Austral. Math. Soc. 75 (2003), 413–422. MR2015326 10.1017/S1446788700008193 1055.46010M. Kato, K.-S. Saito and T. Tamura, On $\psi$-direct sums of Banach spaces and convexity, J. Austral. Math. Soc. 75 (2003), 413–422. MR2015326 10.1017/S1446788700008193 1055.46010
K.-I. Mitani and K.-S. Saito, A note on geometrical properties of Banach spaces using $\psi$-direct sums, J. Math. Anal. Appl. 327 (2007), 898–907. MR2279973 10.1016/j.jmaa.2006.04.059 1109.46017K.-I. Mitani and K.-S. Saito, A note on geometrical properties of Banach spaces using $\psi$-direct sums, J. Math. Anal. Appl. 327 (2007), 898–907. MR2279973 10.1016/j.jmaa.2006.04.059 1109.46017
K.-S. Saito and M. Kato, Uniform convexity of $\psi$-direct sums of Banach spaces, J. Math. Anal. Appl. 277 (2003), 1–11. MR1954459 10.1016/S0022-247X(02)00282-2 1023.46019K.-S. Saito and M. Kato, Uniform convexity of $\psi$-direct sums of Banach spaces, J. Math. Anal. Appl. 277 (2003), 1–11. MR1954459 10.1016/S0022-247X(02)00282-2 1023.46019
K.-S. Saito, M. Kato and Y. Takahashi, Absolute norms on $\mathbb C^n$, J. Math. Anal. Appl. 252 (2000), 879–905. MR1801250 10.1006/jmaa.2000.7139K.-S. Saito, M. Kato and Y. Takahashi, Absolute norms on $\mathbb C^n$, J. Math. Anal. Appl. 252 (2000), 879–905. MR1801250 10.1006/jmaa.2000.7139
M. A. Smith and B. Turett, Rotundity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc. 257 (1980), 105–118. MR549157 10.2307/1998127 0368.46039M. A. Smith and B. Turett, Rotundity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc. 257 (1980), 105–118. MR549157 10.2307/1998127 0368.46039
