This article under its subhead "Embrace prose and avoid terse [read, dense] writing" describes what I mean by "density":
This piece of advice is a reaction against the Bourbaki style ... [of explaining] as little as possible in order to give the tightest presentation possible. ... [I]t is also very hard to read ... an altogether unpleasant experience unless you already know the subject matter and just want to review, not really learn a new subject. ¶ ... Explain ideas fully and clearly. ... [D]o not shy from writing more in order to explain more.
Writing more to explain more is advice unlikely to help a lawyer, who is prolix more often than dense. Writing can be at once prolix and dense, but dense writers usually strive for expository elegance, like the Bourbakis in math. That inverse relationship between prolixity and density — only a trend — shouldn't obscure the different causes of the two mistakes. Nor should the observation that dense writing uses too few words and prolix writing too many. The prolix writer overexplains and the dense writer underexplains, but each is a symptom of a different kind of problem, not the same problem or the opposite one.
Prolixity is actually related to redundancy: prolixity amounts to partial redundancy. A redundant expression repeats identical information; prolix verbiage adds what is practically irrelevant, leaving the reader with nearly identical information. The redundant writer is blind to the repetition, as the prolix writer is to the near repetition. Prolixity comes from a failure of linguistic insight.
Density comes neither from failed linguistic insight nor, of course, its overabundance. Rather, it involves failed psychological insight, in that the dense writer doesn't take the reader's perspective. Density is audience relative: the optimal density for experts is higher than for novices; but density's audience relativity isn't as great as you might think. For an audience of experts, a writer will forgo defining some technical terms, but explanations that organize and activate relevant knowledge help every reader.
(See also related entry Misguided Concision: Terseness.)