The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry

The Lp-Minkowski problem introduced by Lutwak is solved for p⩾n+1 in the smooth category. The relevant Monge–Ampère equation (0.1) is solved for all p>1. The same equation for p<1 is also studied and solved for p∈(-n-1,1). When p=-n-1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan–Warner-type obstruction for this problem is obtained.