On geometric properties of Morrey spaces

The study of Morrey spaces is motivated by many reasons. Initially, these spaces were introduced in order to understand the regularity of solutions to elliptic partial differential equations [1]. In line with this, many authors study the boundedness of various integral operators on Morrey spaces. In this article, we are interested in their geometric properties, from functional analysis point of view. We show constructively that Morrey spaces are not uniformly non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than earlier results, which showed that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{\mathrm{J}}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{\mathrm{NJ}}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le p<q<\infty$ are equal to $n$.