Largest area given a fixed perimeter (Part II)

Over the week, I have found a few more things of note to bring to the metaphorical discussion table. Firstly, there is a most welcome patch to the squabble over the proof of necessity of convexity in maximizing area, and secondly, a proposed sketch of a definition of "inside", used when defining area. However, it is still not clear how the definition can be used explicitly in the proof apart from "obvious"-style handwaving at the moment.

Firstly, the patch:

Consider the following 2 propositions:

1) A larger fixed perimeter will yield a larger maximal area.
2) A straight line is the shortest path between 2 fixed points.

For (2), there is not much to say, and is more or less a take-it-or-leave-it axiomatic definition, due in part to the triangle inequality.

For (1), consider 2 fixed perimeters P and P', with P'>P. Consider the shape with the maximal area for P, S. Define S' similarly. Clearly Area(S')>= Area (similar shape to S with perimeter P')* > Area (S), since Area (S)>0 (consider a square, for example). In fact, Area (similar shape to S with perimeter P') = Area (S) * (P'/P) ^2.

Consider a concave (used here possibly loosely to mean non-convex) shape S. Assume it has maximal area.

Then consider of it, two points on its outline A and B such that there exists a point P on line segment AB such that P is not in the shape.

Next, consider the shape S', which is S with arc AB replaced with line segment AB.

Now, consider the shape S'', which is the union of the set of points contained in S and S'.

The area of S'' is clearly greater than that of S, since there is at least one point contained in S'' not contained in S (P), and whatever is contained in S is by definition contained in S''.

3) The perimeter of S'' is also not more than that of S.

<Proof of (3):

Assume otherwise. If so, then there exists an arc CD in between such that its length is less than line CD. This contradicts (2). We are done.>

Hence, S'' has greater area but less than or equal perimeter to S, and hence a shape S''' similar to S'' but with perimeter equal to S with have greater area than S, contradicting the original assumption that S had maximal area for its perimeter.

Finally, on to part 2 of this post, the definition of "inside":

All points at infinite are outside. Consider any line. Each time it intersects the perimeter of the shape indicates a border between "inside" and "outside". The area of a shape is the sum of areas of all "inside" points.

* Similar as in similar figures, used in the same context as congrurent