The "murders per capita per square mile" metric really gets to the heart of the question on every American's mind: what percentage of face-to-face interactions are murders?

If you wanted to track that you'd need to do the opposite of what GP is suggesting.

That is, a dense urban area would have more face-to-face interactions than a sparse rural one. Given two areas with equal levels of per-capita violence, it would then follow that the more sparsely populated one would have more violence per face-to-face interaction, not less.

Right, I guess you'd actually want a measurement like "murders per population density" or "murders per (person per square mile)". Or, as it's more commonly known, murder-hectares per capita.

Murder-hectares per square capita? Murder-capitas per hectare? I'm not actually sure how the dimensional analysis works out.

This is absolutely absurd. We're talking about the US here, so it would be murder-acres, and that just sounds like a country club you really shouldn't join.

Wouldn't that assume that murderers do some kind of random walk, and randomly kill some person they happen to get close by?

I'm sure there are those, too, but IIRC in most violent crime the attacker and the victim know each other. Do people have substantially bigger social circles in cities?