The Largest Area Same Perimeter

Build rectangles on the side.  

White has actually invested one stone more than Black (19 vs. 18), but Black has 32 points of territory to White's 30. Because the side of the board forms one side of the territory, stones played to increase width count more than stones played to increase height (unless the width is already much greater than the height). Perhaps that's why it's often better to extend along the third or fourth line in the opening and early middle game, rather than building one's moyos upwards. Of course, the fact that territory built higher up is easier to reduce and harder to defend is also important.

1 stone difference  

Bill: In one of his books, Ishida Yoshio repeats the argument in an article in Kido by a mathematician, who claimed, based upon efficient rectangles, that running along the third line (under the opponent's stones on the fourth line) for more than 6 points is inefficient.

My First Attempt  

I believe you can't play at 3,3 and live, although it looks like they could be cut.

My Second Attempt - Circle Theory  

Looks like you could be exposed to double atari. Maybe I need to make the circle bigger?

My Third Attempt  

It's hard to make a circle on a rectangle. I could try to make it bigger, but I think it may result in an increased possibility of a successful invasion at the 3,3 point.

My Fourth Attempt  

Still doesn't look like much of a circle.

Dan: well for any of this the best metric I can think of is (Contained Area)/(stones in perimeter) which gives you area contained per stone which will allow a 1:1 match up between various shapes and a higher number means a better shape up to the point where invasion becomes a major threat for any given shape.