The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere
of ℝn+1. In this paper, we mainly consider the uniqueness question and give first results.
 Panina G., New counterexamples to A.D. Alexandrov’s hypothesis, Adv. Geom., 2005, 5(2), 301–317 http://dx.doi.org/10.1515/advg.2005.5.2.301
 Pogorelov A.V., The Minkowski Multidimensional Problem, Scripta Series in Mathematics, John Wiley & Sons, New York-Toronto-London, 1978
 Schneider R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993 http://dx.doi.org/10.1017/CBO9780511526282
 Stoker J.J., Differential Geometry, Wiley Classics Lib., John Wiley & Sons, New York, 1989
 Zuily C., Existence locale de solutions C
∞ pour des équations de Monge-Ampère changeant de type, Comm. Partial Differential Equations, 1989, 14(6), 691–697 http://dx.doi.org/10.1080/03605308908820627
Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.