Let $X$ be an infinite-dimensional Banach space and $\beta$ denote Istrățescu's spreading measure of noncompactness, i.e. $$\beta(M) = \sup \{ \varepsilon > 0 \colon \exists_{(x_n)^{\mathbb N} \in X^{\mathbb N}} \forall_{m \ne n} \colon \|x_n-x_m\| > \varepsilon \}$$ for any $M \subset X$.

If we denote the (closed) unit ball of $X$ by $B_X$, it is clear that we have $1 \le \beta(B_X) \le 2$ (by Riesz's lemma and the triangle inequality). I believe one has $\beta(B_X) = \sqrt 2$ whenever $X$ is a Hilbert space and $\beta(B_X) = 2$ if $X$ is one of the sequence spaces $c_0$, $\ell^1$, and $\ell^\infty$ or one of the function spaces $L^1[0,1]$ or $L^\infty[0,1]$, so that intuitively, the less round a space, the larger the value $\beta(B_X)$.

**Q**: There is a large zoo of notions of non-roundness, like the (alternative) Daugavet property, the notion of (almost) CL-spaces, lushness, $n(X) = 1$ (where $n$ denotes the numerical index). Is it known how the property $\beta(B_X) = 2$ fits in here? Is it known if this property is inherited by duals or pre-duals? (biduals obviously inherit it). Is it known if such spaces can be reflexive?

**Edit**: It is shown in

> Kryczka, Andrzej; Prus, Stanisław. Separated sequences in nonreflexive Banach spaces. Proc. Amer. Math. Soc. 129 (2001), no. 1, 155--163. [MR1695123](http://www.ams.org/mathscinet-getitem?mr=1695123)

that $\beta(B_X) \ge \sqrt[5]4 \approx 1.3195$ holds for any non-reflexive space $X$.

**Edit**: The following computations show $\beta(B_X) = 2$ for the spaces that were given as examples earlier.

- $X = c_0$ or $X = \ell^\infty$: Consider the sequence $x_1 = (1, 0, \dotsc)$, $x_2 = (-1, 1, 0, \dotsc)$, $x_3 = (-1, -1, 1, \dotsc)$; then $x_m$ and $x_n$ differ by 2 in the coordinate $\min(m,n)$ for $m \ne n$.

- $X = L^\infty[0,1]$: Analogously, consider the sequence $(f_n)$ given by
$$f_n = \begin{cases}
  -1 & \text{on $[0, 1-2^{1-n})$}\\
   1 & \text{on $[1-2^{1-n},1-2^{-n})$}\\
   0 & \text{on $[1-2^{-n},1]$}
\end{cases}$$
for $n \ge 1$. Then $f_n$ and $f_m$ differ by 2 on the interval $[1-2^{1-k},1-2^{-k}]$ with $k = \min(m,n)$ whenever $m \ne n$.

- $X = \ell^1$: Here, it suffices to consider the canonical sequence of unit vectors $x_1 = (1, 0, \dotsc)$, $x_2 = (0, 1, 0, \dotsc)$, $x_3 = (0, 0, 1, 0, \dotsc)$ and so forth (so that $x_n$ and $x_m$ differ by 1 in the $k$-th coordinate for $k \in \{ m, n \}$ whenever $n \ne m$).

- $X = L^1[0,1]$: Analogously, consider the sequence $(f_n)$ given by
$$f_n = \begin{cases}
   0   & \text{on $[0, 1-2^{1-n})$}\\
   2^n & \text{on $[1-2^{1-n},1-2^{-n})$}\\
   0   & \text{on $[1-2^{-n},1]$}
\end{cases}$$
for $n \ge 1$. Then $f_n$ and $f_m$ differ by $2^k$ on the interval $[1-2^{1-k},1-2^{-k}]$ of length $2^{-k}$ for $k \in \{ m, n \}$ whenever $m \ne n$.