In mathematics, there are notations for asymptotic lower bounds, upper bounds, and tight (within-a-constant-factor) bounds. When describing the growth of functions in general, it makes sense that all three might be relevant to some function or other.
In algorithm analysis, asymptotic notation is usually used to describe the growth in time or space requirements.
To me, it seems logical that I may be interested in the maximum, the minimum or the expected requirements. Ignoring the expected requirements, I might be interested in the lower bound of the best case (minimum space/time) or the upper bound of the worst case (maximum space/time).
I can't imagine why I should ever be interested in the lower bound of the worst case or the upper bound of the best case. Since the lower bound is a kind of best case and the upper bound a kind of worst case, these seem like "best case of the worst case" and "worst case of the best case" mismatches to me.
But on several occasions, I've had replies to answers/comments that suggest this may be a failure of imagination on my part.
So - what is it that I'm failing to imagine?
I won't consider a statement along the lines of "it's quite common to say that the lower bound of the worst case of the blah algorithm is blah" to answer the question, unless it also says why people care.