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Martin Sleziak
Martin Sleziak
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I will copy here an exercise from Megginson's book An Introduction to Banach Space Theory, since I think it answers at least partially your question.

For definitions on some notions (and also for some other facts related to this notion) you might want to have a look in this book and the references given there.

Exercise 5.32 (A. L. Garkavi, 1962 [84]). A normed space $X$ is uniformly rotund in every direction or uniformly convex in every direction or directionally uniformly rotund if $\delta_X(\epsilon,\to z)$ whenever $0 < \epsilon < 2$ and $z\in S_X$. The abbreviation URED is used for this property.
(a) Prove that a normed space $X$ is (URED) if and only if it has this property: Whenever $(x_n)$ and $(y_n)$ are sequences in $S_X$ such that $\|\frac12(x_n+y_n)\|\to1$ and such that $x_n-t_n\in\langle\{v\}\rangle$ for some $v$ in $X$ and each $n$, it follows that $x_n — y_n \to 0$.
(b) Show that (wUR) $\Rightarrow$ (URED) $\Rightarrow$ (R).
(c) (This uses material from Exercise 5.31). Show that (URWC) $\Rightarrow$ (URED).

Smith gave an example in [219] of a Banach space that is (URED) but not (URWC), and another Banach space that is (R) but not (URED); both examples are formed by equivalently renorming $\ell_2$. It can be shown that a normed space $X$ is (URED) if and only if $\delta_X(\epsilon,\to A) > 0$ whenever $0 < \epsilon < 2$ and $A$ is a nonempty compact subset of $X\setminus\{0\}$; see [218]. Another good source of information on uniform rotundity in every direction is the paper of Day, James, and Swaminathan [57] devoted to the property.

Since uniformly rotund implies weakly uniformly rotund, the part (b) of this exercise answers at least the question about relation of (UR) and (URED). We get that (UR)$\Rightarrow$(URED). Smith's example should be a counterexample for the opposite implication. ((URWC) is defined in the previous exercise, and it fulfills (wUR)$\Rightarrow$(URWC)$\Rightarrow$(R).

References mentioned in this excerpt:

  • [57] Mahlon M. Day, Robert C. James, and Srinivasa Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059, http://dx.doi.org/10.4153/CJM-1971-109-5
  • [84] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106 (Russian, translated into English as [85]);
  • [85] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl., Ser. 2 39 (1964), 111-132 (English translation of [84]).
  • [218] Mark A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canad. J. Math. 29 (1977), 963-970; http://dx.doi.org/10.4153/CJM-1977-097-6
  • [219] Mark A. Smith, Some examples concerning rotundity in Banach spaces, Mathematische Annalen 1978, Volume 233, Issue 2, pp 155-161, http://dx.doi.org/10.1007/BF01421923

I will copy here an exercise from Megginson's book An Introduction to Banach Space Theory, since I think it answers at least partially your question.

For definitions on some notions (and also for some other facts related to this notion) you might want to have a look in this book and the references given there.

Exercise 5.32 (A. L. Garkavi, 1962 [84]). A normed space $X$ is uniformly rotund in every direction or uniformly convex in every direction or directionally uniformly rotund if $\delta_X(\epsilon,\to z)$ whenever $0 < \epsilon < 2$ and $z\in S_X$. The abbreviation URED is used for this property.
(a) Prove that a normed space $X$ is (URED) if and only if it has this property: Whenever $(x_n)$ and $(y_n)$ are sequences in $S_X$ such that $\|\frac12(x_n+y_n)\|\to1$ and such that $x_n-t_n\in\langle\{v\}\rangle$ for some $v$ in $X$ and each $n$, it follows that $x_n — y_n \to 0$.
(b) Show that (wUR) $\Rightarrow$ (URED) $\Rightarrow$ (R).
(c) (This uses material from Exercise 5.31). Show that (URWC) $\Rightarrow$ (URED).

Smith gave an example in [219] of a Banach space that is (URED) but not (URWC), and another Banach space that is (R) but not (URED); both examples are formed by equivalently renorming $\ell_2$. It can be shown that a normed space $X$ is (URED) if and only if $\delta_X(\epsilon,\to A) > 0$ whenever $0 < \epsilon < 2$ and $A$ is a nonempty compact subset of $X\setminus\{0\}$; see [218]. Another good source of information on uniform rotundity in every direction is the paper of Day, James, and Swaminathan [57] devoted to the property.

Since uniformly rotund implies weakly uniformly rotund, the part (b) of this exercise answers at least the question about relation of (UR) and (URED). We get that (UR)$\Rightarrow$(URED). Smith's example should be a counterexample for the opposite implication. ((URWC) is defined in the previous exercise, and it fulfills (wUR)$\Rightarrow$(URWC)$\Rightarrow$(R).

References mentioned in this excerpt:

  • [57] Mahlon M. Day, Robert C. James, and Srinivasa Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059, http://dx.doi.org/10.4153/CJM-1971-109-5
  • [84] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106 (Russian, translated into English as [85]);
  • [85] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl., Ser. 2 39 (1964), 111-132 (English translation of [84]).
  • [218] Mark A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canad. J. Math. 29 (1977), 963-970; http://dx.doi.org/10.4153/CJM-1977-097-6

I will copy here an exercise from Megginson's book An Introduction to Banach Space Theory, since I think it answers at least partially your question.

For definitions on some notions (and also for some other facts related to this notion) you might want to have a look in this book and the references given there.

Exercise 5.32 (A. L. Garkavi, 1962 [84]). A normed space $X$ is uniformly rotund in every direction or uniformly convex in every direction or directionally uniformly rotund if $\delta_X(\epsilon,\to z)$ whenever $0 < \epsilon < 2$ and $z\in S_X$. The abbreviation URED is used for this property.
(a) Prove that a normed space $X$ is (URED) if and only if it has this property: Whenever $(x_n)$ and $(y_n)$ are sequences in $S_X$ such that $\|\frac12(x_n+y_n)\|\to1$ and such that $x_n-t_n\in\langle\{v\}\rangle$ for some $v$ in $X$ and each $n$, it follows that $x_n — y_n \to 0$.
(b) Show that (wUR) $\Rightarrow$ (URED) $\Rightarrow$ (R).
(c) (This uses material from Exercise 5.31). Show that (URWC) $\Rightarrow$ (URED).

Smith gave an example in [219] of a Banach space that is (URED) but not (URWC), and another Banach space that is (R) but not (URED); both examples are formed by equivalently renorming $\ell_2$. It can be shown that a normed space $X$ is (URED) if and only if $\delta_X(\epsilon,\to A) > 0$ whenever $0 < \epsilon < 2$ and $A$ is a nonempty compact subset of $X\setminus\{0\}$; see [218]. Another good source of information on uniform rotundity in every direction is the paper of Day, James, and Swaminathan [57] devoted to the property.

Since uniformly rotund implies weakly uniformly rotund, the part (b) of this exercise answers at least the question about relation of (UR) and (URED). We get that (UR)$\Rightarrow$(URED). Smith's example should be a counterexample for the opposite implication. ((URWC) is defined in the previous exercise, and it fulfills (wUR)$\Rightarrow$(URWC)$\Rightarrow$(R).

References mentioned in this excerpt:

  • [57] Mahlon M. Day, Robert C. James, and Srinivasa Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059, http://dx.doi.org/10.4153/CJM-1971-109-5
  • [84] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106 (Russian, translated into English as [85]);
  • [85] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl., Ser. 2 39 (1964), 111-132 (English translation of [84]).
  • [218] Mark A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canad. J. Math. 29 (1977), 963-970; http://dx.doi.org/10.4153/CJM-1977-097-6
  • [219] Mark A. Smith, Some examples concerning rotundity in Banach spaces, Mathematische Annalen 1978, Volume 233, Issue 2, pp 155-161, http://dx.doi.org/10.1007/BF01421923
Source Link
Martin Sleziak
Martin Sleziak
  • 4.7k
  • 4
  • 35
  • 40

I will copy here an exercise from Megginson's book An Introduction to Banach Space Theory, since I think it answers at least partially your question.

For definitions on some notions (and also for some other facts related to this notion) you might want to have a look in this book and the references given there.

Exercise 5.32 (A. L. Garkavi, 1962 [84]). A normed space $X$ is uniformly rotund in every direction or uniformly convex in every direction or directionally uniformly rotund if $\delta_X(\epsilon,\to z)$ whenever $0 < \epsilon < 2$ and $z\in S_X$. The abbreviation URED is used for this property.
(a) Prove that a normed space $X$ is (URED) if and only if it has this property: Whenever $(x_n)$ and $(y_n)$ are sequences in $S_X$ such that $\|\frac12(x_n+y_n)\|\to1$ and such that $x_n-t_n\in\langle\{v\}\rangle$ for some $v$ in $X$ and each $n$, it follows that $x_n — y_n \to 0$.
(b) Show that (wUR) $\Rightarrow$ (URED) $\Rightarrow$ (R).
(c) (This uses material from Exercise 5.31). Show that (URWC) $\Rightarrow$ (URED).

Smith gave an example in [219] of a Banach space that is (URED) but not (URWC), and another Banach space that is (R) but not (URED); both examples are formed by equivalently renorming $\ell_2$. It can be shown that a normed space $X$ is (URED) if and only if $\delta_X(\epsilon,\to A) > 0$ whenever $0 < \epsilon < 2$ and $A$ is a nonempty compact subset of $X\setminus\{0\}$; see [218]. Another good source of information on uniform rotundity in every direction is the paper of Day, James, and Swaminathan [57] devoted to the property.

Since uniformly rotund implies weakly uniformly rotund, the part (b) of this exercise answers at least the question about relation of (UR) and (URED). We get that (UR)$\Rightarrow$(URED). Smith's example should be a counterexample for the opposite implication. ((URWC) is defined in the previous exercise, and it fulfills (wUR)$\Rightarrow$(URWC)$\Rightarrow$(R).

References mentioned in this excerpt:

  • [57] Mahlon M. Day, Robert C. James, and Srinivasa Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059, http://dx.doi.org/10.4153/CJM-1971-109-5
  • [84] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106 (Russian, translated into English as [85]);
  • [85] Aleksandr L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc. Transl., Ser. 2 39 (1964), 111-132 (English translation of [84]).
  • [218] Mark A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canad. J. Math. 29 (1977), 963-970; http://dx.doi.org/10.4153/CJM-1977-097-6
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