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The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $\|\cdot\|_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto\|x\|_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $\|\cdot\|_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

The answer is provided by a combination of:

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),\|\cdot\|_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

This means, in particular, that the following assumptions (included in the original post) are obsolete:

1. $\Phi$ is an N-function (i.e. $\lim_{u\rightarrow^+0}\frac{\Phi(u)}{u}=0$ and $\lim_{u\rightarrow\infty}\frac{\Phi(u)}{u}=\infty$);
2. strengthening of $\Delta_2^\infty$ to $\Delta_2$ (i.e. assuming also that $\lim_{u\rightarrow^+0}\frac{\Phi(2u)}{\Phi(u)}<\infty$).

References

[1] Anna Kamińska, 1984, "The criteria for local uniform rotundity of Orlicz spaces", Stud. Math. 79, 201–215, MR0781718, Zbl 0573.46014.

[2] Rudolf Výborný V., 1956, "O slabé konvergenci v prostorech lokálně stejnoměrně konvexních", Časopis pěstov. matem. 81, 352–353, Zbl 0075.11702.

[3] Vitol'd L'vovich Shmul'yan, 1939, "On some geometrical properties of the sphere in a space of the type (B)", Dokl. Akad. nauk SSSR 24, 648–652, .

[4] Marston Morse, William Transue, 1950, "Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$", Ann. Math. 51, 576–614, MR31654, Zbl 0035.07105.

[5] Hidegorô Nakano (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō, MR38565, Zbl 0041.23401.

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $\|\cdot\|_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto\|x\|_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $\|\cdot\|_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

The answer is provided by a combination of:

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),\|\cdot\|_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

This means, in particular, that the following assumptions (included in the original post) are obsolete:

1. $\Phi$ is an N-function (i.e. $\lim_{u\rightarrow^+0}\frac{\Phi(u)}{u}=0$ and $\lim_{u\rightarrow\infty}\frac{\Phi(u)}{u}=\infty$);
2. strengthening of $\Delta_2^\infty$ to $\Delta_2$ (i.e. assuming also that $\lim_{u\rightarrow^+0}\frac{\Phi(2u)}{\Phi(u)}<\infty$).

References

[1] Anna Kamińska, 1984, "The criteria for local uniform rotundity of Orlicz spaces", Stud. Math. 79, 201–215, MR0781718, Zbl 0573.46014.

[2] Rudolf Výborný, 1956, "O slabé konvergenci v prostorech lokálně stejnoměrně konvexních", Časopis pěstov. matem. 81, 352–353, Zbl 0075.11702.

[3] Vitol'd L'vovich Shmul'yan, 1939, "On some geometrical properties of the sphere in a space of the type (B)", Dokl. Akad. nauk SSSR 24, 648–652.

[4] Marston Morse, William Transue, 1950, "Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$", Ann. Math. 51, 576–614, MR31654, Zbl 0035.07105.

[5] Hidegorō Nakano (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō, MR38565, Zbl 0041.23401.

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $||\cdot||_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto||x||_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $||\cdot||_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),||\cdot||_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

References:

[1] Kamińska A., 1984, The criteria for local uniform rotundity of Orlicz spaces, Stud. Math. 79, 201–215. http://matwbn.icm.edu.pl/ksiazki/sm/sm79/sm79117.pdf.

[2] Výborný V., 1956, O slabé konvergenci v prostorech lokálně stejnoměrně konvexních, Časopis pěstov. matem. 81, 352–353. https://dml.cz/bitstream/handle/10338.dmlcz/117195/CasPestMat_081-1956-3_10.pdf.

[3] Shmul'yan V.L., 1939, On some geometrical properties of the sphere in a space of the type (B), Dokl. Akad. nauk SSSR 24, 648–652. https://www.fuw.edu.pl/~kostecki/scans/shmulyan1939.pdf.

[4] Morse M., Transue W., 1950, Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$, Ann. Math. 51, 576–614.

[5] Nakano H. (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō.

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $\|\cdot\|_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto\|x\|_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $\|\cdot\|_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),\|\cdot\|_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

References

[1] Anna Kamińska, 1984, "The criteria for local uniform rotundity of Orlicz spaces", Stud. Math. 79, 201–215, MR0781718, Zbl 0573.46014.

[2] Rudolf Výborný V., 1956, "O slabé konvergenci v prostorech lokálně stejnoměrně konvexních", Časopis pěstov. matem. 81, 352–353, Zbl 0075.11702.

[3] Vitol'd L'vovich Shmul'yan, 1939, "On some geometrical properties of the sphere in a space of the type (B)", Dokl. Akad. nauk SSSR 24, 648–652, .

[4] Marston Morse, William Transue, 1950, "Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$", Ann. Math. 51, 576–614, MR31654, Zbl 0035.07105.

[5] Hidegorô Nakano (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō, MR38565, Zbl 0041.23401.

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $||\cdot||_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto||x||_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $||\cdot||_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

The answer is provided by a combination of:

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),||\cdot||_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

This means, in particular, that the following assumptions (included in the original post) are obsolete:

1. $\Phi$ is an N-function (i.e. $\lim_{u\rightarrow^+0}\frac{\Phi(u)}{u}=0$ and $\lim_{u\rightarrow\infty}\frac{\Phi(u)}{u}=\infty$);
2. strengthening of $\Delta_2^\infty$ to $\Delta_2$ (i.e. assuming also that $\lim_{u\rightarrow^+0}\frac{\Phi(2u)}{\Phi(u)}<\infty$).

[1] Kamińska A., 1984, The criteria for local uniform rotundity of Orlicz spaces, Stud. Math. 79, 201–215. http://matwbn.icm.edu.pl/ksiazki/sm/sm79/sm79117.pdf.

[2] Výborný V., 1956, O slabé konvergenci v prostorech lokálně stejnoměrně konvexních, Časopis pěstov. matem. 81, 352–353. https://dml.cz/bitstream/handle/10338.dmlcz/117195/CasPestMat_081-1956-3_10.pdf.

[3] Shmul'yan V.L., 1939, On some geometrical properties of the sphere in a space of the type (B), Dokl. Akad. nauk SSSR 24, 648–652. https://www.fuw.edu.pl/~kostecki/scans/shmulyan1939.pdf.

[4] Morse M., Transue W., 1950, Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$, Ann. Math. 51, 576–614.

[5] Nakano H. (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō.

The question essentially boils down to asking about sufficient conditions under which an Orlicz space $L_\Phi(\mathcal{X},\mu)$ over a nonatomic finite measure space $(\mathcal{X},\mu)$, equipped with the Morse–Transue–Nakano norm $||\cdot||_\Phi$, $$ L_\Phi(\mathcal{X},\mu)\ni x\mapsto||x||_\Phi:=\inf\left\{\lambda>0\mid \int_{\mathcal{X}}\mu\,\Phi\left(\frac{x}{\lambda}\right)\leq1\right\}\in\mathbb{R}^+, $$ has the Radon–Riesz–Shmul'yan property.

(A historical side comment: it was Shmul'yan [3, Thm. 5] who first introduced this property for general Banach spaces. Kadec (in 1958) cites this paper, while Klee (in 1960) cites Kadec, so "Kadec–Klee property" is a misnomer. (It is possible that Výborný [2, Prop. (p. 352)], who is also cited in the same paper of Kadec, has introduced this property independently of Shmul'yan.) Furthermore, while $||\cdot||_\Phi$ is usually called "Luxemburg norm", this is also a misnomer: Luxemburg (in 1955) actually cites the book Nakano, where this norm was introduced [5, p. 181]. Morse and Transue introduced it in [4, Def. (p. 610)], independently from Nakano.)

The answer is provided by a combination of:

1. [1, Thm. 1]: for any finite Young function (meaning: $\Phi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is convex, with $\Phi(0)=0$ and $\Phi\not\equiv0$), an Orlicz space $(L_\Phi(\mathcal{X},\mu),||\cdot||_\Phi)$, over a nonatomic finite measure space $(\mathcal{X},\mu)$, is locally uniformly convex if and only if $\Phi$ is strictly convex on $\mathbb{R}^+$ and satisfies $\Delta_2^\infty$ condition (i.e. $\limsup_{u\rightarrow\infty}\frac{\Phi(2u)}{\Phi(u)}<\infty$);
2. [2, Prop. (p. 352)]: if a Banach space is locally uniformly convex, then it has the Radon–Riesz–Shmul'yan property.

This means, in particular, that the following assumptions (included in the original post) are obsolete:

1. $\Phi$ is an N-function (i.e. $\lim_{u\rightarrow^+0}\frac{\Phi(u)}{u}=0$ and $\lim_{u\rightarrow\infty}\frac{\Phi(u)}{u}=\infty$);
2. strengthening of $\Delta_2^\infty$ to $\Delta_2$ (i.e. assuming also that $\lim_{u\rightarrow^+0}\frac{\Phi(2u)}{\Phi(u)}<\infty$).

References:

[1] Kamińska A., 1984, The criteria for local uniform rotundity of Orlicz spaces, Stud. Math. 79, 201–215. http://matwbn.icm.edu.pl/ksiazki/sm/sm79/sm79117.pdf.

[2] Výborný V., 1956, O slabé konvergenci v prostorech lokálně stejnoměrně konvexních, Časopis pěstov. matem. 81, 352–353. https://dml.cz/bitstream/handle/10338.dmlcz/117195/CasPestMat_081-1956-3_10.pdf.

[3] Shmul'yan V.L., 1939, On some geometrical properties of the sphere in a space of the type (B), Dokl. Akad. nauk SSSR 24, 648–652. https://www.fuw.edu.pl/~kostecki/scans/shmulyan1939.pdf.

[4] Morse M., Transue W., 1950, Functionals $F$ bilinear over the product $A$ $\times$ $B$ of two pseudo-normed vector spaces II. Admissible spaces $A$, Ann. Math. 51, 576–614.

[5] Nakano H. (中野 秀五郎), 1950, Modulared semi-ordered linear spaces, Maruzen, Tōkyō.