Functions in $L_1(\mu,Y)$ with optimal tensor representations

We study the existence and characterization of optimal tensor representations of elements in the space $L_1(\mu,Y)$ of Bochner integrable functions.

We completely describe the set of norm-attaining elements in two settings.

First, when the Banach space $Y$ is strictly convex, and second, when $Y=L_1(\nu)$ and $\mathbb K=\mathbb R$.

In both situations, our analysis yields the existence of non-norm-attaining tensors whenever the underlying measures are not purely atomic.

Finally, we introduce a geometric property over $Y$ ensuring that every element in $L_1(\mu, Y)$ admits an optimal representation.

In particular, this holds for Lipschitz-free spaces over complete scattered metric spaces, for $C(K)$ spaces when $K$ is a compact Hausdorff totally disconnected space, and for $c_0(\Gamma)$ where $\Gamma$ is any index set.

As a byproduct, we settle two open questions regarding projective norm-attainment.