Suppose you are going to run N miles in a single, contiguous run in “virgin” territory (i.e. no completed streets anywhere near you).
What is the largest territory you can create in that single run, and what is the shape of the path you should run to achieve it?
(You can assume that there is a street that lies exactly along the mathematical path you want to take.)
My guess: Two sides of an equidistant square, like a capital L, each side 0.5N in length?
The largest area for a given perimeter is a circle (proof: https://math.stackexchange.com/questions/461853/given-a-fixed-perimeter-which-shape-will-have-the-maximum-area#1123846) therefore my guess is running a giant circular arc, the largest that would fit on any given continent (or two?)
But maybe we need to take into consideration we are running on the surface of a sphere?
I was assuming that the N-mile run I specified in the problem was of a more mundane length, so let’s neglect continental-length runs and Earth’s oblate spheroid shape. Flat Earth Society for this one. 
When you say “a giant circular arc”, that is not quite a full specification of the solution. (The optimal solution is not a circle, unless one forces an additional assumption on the problem that I did not state.)
I reckon it’s a semicircular run, the area then would be N²/2π sq. miles.
Yes, I assumed the “complete” peak area for mileage would be up to a half-circle, of whatever radius you could fit.
Related https://arxiv.org/pdf/1804.07389
“Longest Straight Line Paths on Water or Land on the Earth”
Spoiler:
Alternate view:
