I am looking for a good data structure for storing a set of points $P\subset \mathbb{N}^n$ that is able to answer the following query:
Given a point $x=(x_1,\cdots,x_n)$, does there exist a point $p = (p_1,\cdots,p_n)\in P$ such that $p \leq x$, meaning that $p_i\leq x_i$ for all $i\in \{1,\cdots, n\}$?
If the answer is yes, then the data structure should return one such point.
Ideally, it should be possible to dynamically add points to $P$.
The data structure can 'forget' a point $p \in P$ if there exists another point $p'$ with $p' \leq p$.
The use case that I am thinking about has 'small points'. Something like $x = (x_i)$ with $\sum_i x_i \leq 30$. But has a somewhat larger $n$.
ps: I have removed the copy on SO.