I'm sure the solution is known for the 2D problem, but I can't find a source for the solution at the moment. I'm fairly sure the maximum achieved by the 2D analogue of the conjectured set, but not totally sure.

Elaborating on JBL: This can't be determined from just what is given (which is essentially no information about $K$). Could you state what information is known about the body in question? For instance, do we know it's support function?

This seems likely to get closed, but before it does, I'd like to point out that a decent textbook accomplishes all of these except being software. Is there any reason why you feel a textbook is not sufficient to learn topology?

Excellent! The answer is both easy to understand and to verify. I'm a bit upset I couldn't think of it myself.

+1: This one came to my mind as well. Not necessarily the situation described in the OP, since the original "proof" had a number of holes, and the shorter one almost fell out from the results used to fill the gaps. It does show how math can be drastically simplified by the right approach, which captures the spirit of the question, if not the letter.

Thanks! I tried to no avail to embed the image, but I'm still rather bad with TeX and couldn't get it to work.