Opalg, I did undserstand what you wrote. The only part I'm not quite sure what you meant was the end.

By writing : "and hence (by taking scalar multiples) on the whole of

"

did you mean that for every (x,y) != (0,0) you can divide the term

|XZ1+yZ2| by |x|+|y| and therfore to get that x' = x/(|x|+|y|) and

y' = y/(|x|+|y|) will obtain : |x'| +|y'| = 1 and (x',y') belong to S (and from there the original inequality derives ?).

If it was not what you meant I'll be glad to get you short clarification on another post. Thank you cery much !