Offline multidimensional RMQ/RSQ in query model Problem: In the multidimensional range Max/Sum query problem (RMQ/RSQ)
  you are given a $d$-dimensional array
  with $n$ elements, and given a
  $d$-dimensional box, you wish to
  determine the max/sum of all elements
  in the range (if the elements are from
  semigroup, they have no inverses).


What is the state-of-art best algorithm for offline multidimensional RMQ/RSQ in query model (preprocessing of data allowed in offline, queries asked online)? Is there any gap between RMQ and RSQ in multidimensional case? What theoretical/practical interest does this problem have, if any?

I'm interested in it because looks like the best asymptotics for both RMQ and RSQ was achieved in this paper by Chazelle and Rosenberg in 1989, which was not cited a lot, though the problem looks very natural and interesting. By best asymptotics I mean $O(\alpha^d(m,n))$ query time, where $m$ is the amount of storage available per element, $d$ --- number of dimensions of the array, $\alpha$ --- inverse Ackerman function, $n$ --- number of entries in the array. I suspect that this is mainly because this paper is not very readable in the part where it describes the construction, so implementation of this approach is complicated. So a simpler description of a data structure with the same asymptotics is also very welcome.
So, you are interested in range queries in the "dense case" (an array) as opposed to the case most frequently studied in theory (arbitrary points in space).

The RMQ problem is easier than the semigroup sum problem, and no $\alpha$ is needed. The most recent result I know is: Hao Yuan, Mikhail J. Atallah: Data Structures for Range Minimum Queries in Multidimensional Arrays, SODA 2010

The semigroup sum problem is mainly of theoretical interest, in my opinion (what is a natural semigroup beyond max?). This explains why the Chazelle-Rosenberg paper does not have a lot of citations.