It is proved that the solution of the multiplicative Cauchy functional equation on the Lorentz cone of dimension greater than two is a power function of the determinant. The equation is solved in full generality, i.e. no smoothness assumptions on the unknown function are imposed. Also the functional equation of ratios, of a similar nature, is solved in full generality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-3-4,
author = {Jacek Weso\l owski},
title = {Multiplicative Cauchy functional equation and the equation of ratios on the Lorentz cone},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {263-275},
zbl = {1114.39009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-3-4}
}
Jacek Wesołowski. Multiplicative Cauchy functional equation and the equation of ratios on the Lorentz cone. Studia Mathematica, Tome 178 (2007) pp. 263-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-3-4/