Abstract
The L p -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.
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