On the construction of self-polar and self-polar Hilbertian norms on Minkowski space of dimension 2

Considering the real Minkowski space M-2 endowed with the indefinite inner product ((a) over right arrow,(b) over right arrow)=x(a)x(b)-y(a)y(b), (a) over right arrow=(x(a),y(a)), (b) over right arrow=(x(b),y(b))is an element ofM(2), we give a general construction for the boundary partial derivativeU(p)={(a) over right arrow is an element ofM(2); p((a) over right arrow)=1} of the unit ball U-p of some self-polar norm p on M-2. We show further that every self-polar norm p on M-2 is obtained by our construction. A detailed investigation of the possible choices of the ingredients of our construction (two points P-j, j=1,2, and a convex arc connecting P-1 and P-2) yields a complete classification of all self-polar norms on M-2. Our results are finally applied to the special case of Hilbertian self-polar norms. (C) 2003 American Institute of Physics.