The maximum of minimum angle in multi-dimensional Euclidean space

Question

I'm interested in upper bounds of the following value: \begin{align} \max_{{\bf x}_1,\ldots,{\bf x}_k \in \mathbb{R}^n \setminus \{{\bf 0}\}} \min_{i\neq j} \frac{\langle{\bf x}_i,{\bf x}_j\rangle}{|{\bf x}_i||{\bf x}_j|}. \end{align} I think that this max-min problem is related to multi-dimensional packing problem but I don't know what the exact keyword is. Please recommend some keywords or links. Thanks.