The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry.(English)Zbl 1245.52001
Summary: The \(L_p\)-Minkowski problem introduced by Lutwak is solved for \(p\geq n+1\) in the smooth category. The relevant Monge-Ampère equation \[\det(h_{ij}+h\delta_{ij})=fh^{p-1} \tag{0.1}\] is solved for all \(p>1\) where \(h_{ij}\) denotes convariant differentiation. The same equation for \(p<1\) is also studied and solved for \(p\in(-n-1,1)\). 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.
MSC:
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |
| 52A39 | Mixed volumes and related topics in convex geometry |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 53A15 | Affine differential geometry |
Keywords:
\(L_p\)-Minkowski problem;centroaffine Minkowski problem;Monge-Ampère equation;Blaschke-Santalo inequalityCite
Full Text:DOI
References:
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