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

Abstract The L p -Minkowski problem introduced by Lutwak is solved for p ⩾ n + 1 in the smooth category. The relevant Monge–Ampere 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.

[1]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

[2]  E. Lutwak,et al.  Sharp Affine LP Sobolev Inequalities , 2002 .

[3]  Xu-jia Wang,et al.  A logarithmic Gauss curvature flow and the Minkowski problem , 2000 .

[4]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[5]  B. Andrews,et al.  Motion of hypersurfaces by Gauss curvature , 2000 .

[6]  L. Nirenberg,et al.  Partial Differential Equations Invariant under Conformal or Projective Transformations , 1974 .

[7]  Ben Andrews,et al.  Evolving convex curves , 1998 .

[8]  Juncheng Wei,et al.  Self-similar solutions for the anisotropic affine curve shortening problem , 2001 .

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  B. Andrews.  Classification of limiting shapes for isotropic curve flows , 2002 .

[11]  L. Caffarelli.  Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation , 1990 .

[12]  Erwin Lutwak,et al.  The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem , 1993 .

[13]  E. Lutwak,et al.  On the _{}-Minkowski problem , 2003 .

[14]  Wm. J. Firey,et al.  $p$-Means of Convex Bodies. , 1962 .

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Luis A. Caffarelli,et al.  A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity , 1990 .

[17]  J. Urbas.  Complete noncompact self-similar solutions of Gauss curvature flows I. Positive powers , 1998, Advances in Differential Equations.

[18]  B. Andrews.  Gauss curvature flow: the fate of the rolling stones , 1999 .

[19]  M. Georges Tzitzéica,et al.  Sur une nouvelle classe de surfaces , 1908 .

[20]  Xu-jia Wang,et al.  A variational theory of the Hessian equation , 2001 .

[21]  Shing-Tung Yau,et al.  On the regularity of the monge‐ampère equation det (∂2 u/∂xi ∂xj) = f(x, u) , 1977 .

[22]  V. Oliker,et al.  On the regularity of solutions to a generalization of the Minkowski problem , 1995 .