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If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x_0$ at which the condition of this definition is satisfied, has the negation of the property described in your Question 1. The converse does not seem to be true, as it seems that you need more than the negation of local uniform convexity, you need a kind of symmetric lack of the local uniform convexity.

As an example of a strictly convex Banach space without the property described in Question 1 I suggest the space $c_0$ with the norm $$||\{a_i\}_{i=1}^\infty||=\max_i|a_i|+\left(\sum_{i=1}^\infty |a_i|^2/2^i\right)^{1/2}.$$

If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x_0$ at which the condition of this definition is satisfied, has the property described in your Question 1. The converse does not seem to be true, as it seems that you need more than the negation of local uniform convexity, you need a kind of symmetric lack of the local uniform convexity.

As an example of a strictly convex Banach space without the property described in Question 1 I suggest the space $c_0$ with the norm $$||\{a_i\}_{i=1}^\infty||=\max_i|a_i|+\left(\sum_{i=1}^\infty |a_i|^2/2^i\right)^{1/2}.$$

If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x_0$ at which the condition of this definition is satisfied, has the property described in your Question 1. The converse does not seem to be true, as it seems that you need more than the negation of local uniform convexity, you need a kind of symmetric lack of the local uniform convexity.

As an example of a strictly convex Banach space without the property described in Question 1 I suggest the space $c_0$ with the norm $$||\{a_i\}_{i=1}^\infty||=\max_i|a_i|+\left(\sum_{i=1}^\infty |a_i|^2/2^i\right)^{1/2}.$$

If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x_0$ at which the condition of this definition is satisfied, has the negation of the property described in your Question 1. The converse does not seem to be true, as it seems that you need more than the negation of local uniform convexity, you need a kind of symmetric lack of the local uniform convexity.

As an example of a strictly convex Banach space without the property described in Question 1 I suggest the space $c_0$ with the norm $$||\{a_i\}_{i=1}^\infty||=\max_i|a_i|+\left(\sum_{i=1}^\infty |a_i|^2/2^i\right)^{1/2}.$$

If you look at the definition of (strongly) locally uniformly convex space in Diestel, Geometry of Banach spaces, Section 2 of Chapter II, you will see that this is what is needed for the desired property to hold. Strict convexity is not enough. And there are examples of strictly convex Banach spaces which are not locally uniformly convex, you can even re-norm $\ell_2$ to achieve this.

If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x_0$ at which the condition of this definition is satisfied, has the property described in your Question 1. The converse does not seem to be true, as it seems that you need more than the negation of local uniform convexity, you need a kind of symmetric lack of the local uniform convexity.

As an example of a strictly convex Banach space without the property described in Question 1 I suggest the space $c_0$ with the norm $$||\{a_i\}_{i=1}^\infty||=\max_i|a_i|+\left(\sum_{i=1}^\infty |a_i|^2/2^i\right)^{1/2}.$$