Proximinality in Banach Space-Valued Grand Bochner–Lebesgue Spaces with Variable Exponent

Let $(A,\mathscr{A},\mu)$ be a $\sigma$-finite complete measure space, and let $p(\cdot)$ be a $\mu$-measurable function on $A$ which takes values in $(1,\infty)$. Let $Y$ be a subspace of a Banach space $X$. By $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ and $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ we denote the grand Bochner–Lebesgue spaces with variable exponent $p(\cdot)$ whose functions take values in $Y$ and $X$, respectively. First, we estimate the distance of $f$ from $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ when $f\in \widetilde{L}^{p(\cdot),\varphi}(A, X)$. Then we prove that $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ is proximinal in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ if $Y$ is weakly $\mathcal{K}$-analytic and proximinal in $X$. Finally, we establish a connection between the proximinality of $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ and the proximinality of $L^1(A, Y)$ in $L^1(A, X)$.