  In this article, the authors completely answer an open question, presented in
[Banach J. Math. Anal. 15 (2021), no. 1, 20], via showing that the
Riesz--Morrey space is truly a new space larger than a particular Lebesgue
space with critical index. Indeed, this Lebesgue space is just the real
interpolation space of the Riesz--Morrey space for suitable indices. Moreover,
the authors further show the aforementioned inclusion is also proper, namely,
this embedding is sharp in some sense, via constructing two nontrivial spare
functions, respectively, on $\mathbb{R}^n$ and any given cube $Q_0$ of
$\mathbb{R}^n$ with finite side length. The latter constructed function is
inspired by the striking function constructed by Dafni et al. [J. Funct. Anal.
275 (2018), 577--603]. All the proofs of these results strongly depend on some
exquisite geometrical analysis on cubes of $\mathbb{R}^n$. As an application,
the relationship between Riesz--Morrey spaces and Lebesgue spaces is completely
clarified on all indices.
