= Solution
Sort the two dimension vectors separately into ascending order, $a_1\le\cdots\le a_d$ and $b_1\le\cdots\le b_d$. Then return \b[true exactly when $a_i<b_i$ for every $i$]. Sorting costs $O(d\log d)$ and the coordinate comparisons cost $O(d)$.
Sufficiency follows by matching corresponding ranks and translating that rank matching back to the original axes. For necessity, suppose $a_k\ge b_k$ for some $k$. The $d-k+1$ dimensions $a_k,\ldots,a_d$ must all be assigned to strictly larger target dimensions. None of the first $k$ sorted target dimensions is large enough, leaving at most $d-k$ eligible slots. The <pigeonhole principle> gives a contradiction. This proves the <sorted-coordinate criterion for box containment>, including ties: equality fails the required strict fit.
Back to article page
