Let $\mathcal{P} := \{ x_1, x_2, \ldots, x_k \} \subset \mathbb{R}^d$ and let $\mathcal{C}$ be the convex hull of $\mathcal{P}$. What is the minimal Euclidean distance between $q \in \mathbb{R}^d$ and any point in $\mathcal{C}$?
I am more interested in cases where $ 2 \leq k \leq 100$ and $d \approx 1000$. It is also argued that the Euclidean distance might be non-intuitive in high dimensions. In that case, what other distance functions could be considered?