Notation for choosing the minimum of a set after applying the norm

Question

I have a function $x: \mathbb{R^+} \mapsto \mathbb{R}^n$ defined as $x(k) = \text{min}\{p \in S: \lVert p \rVert > k\}$ (where $S \subseteq \mathbb{R}^n$). I want to choose the point with the smallest norm larger than $k$, but I feel that I should be describing this in 'min' somehow. Is there any notation to describe this or is it best I just do it with words? Assume the set is always non-empty.