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

@article{Chou2006TheLP, title={The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry}, author={Kai-Seng Chou and Xu-jia Wang}, journal={Advances in Mathematics}, year={2006}, volume={205}, pages={33-83} }

## 254 Citations

The $L_p$-Minkowski problem with super-critical exponents

- Mathematics
- 2022

The Lp-Minkowski problem deals with the existence of closed convex hypersurfaces in R with prescribed p-area measures. It extends the classical Minkowski problem and embraces several important…

On the discrete Orlicz Minkowski problem

- MathematicsTransactions of the American Mathematical Society
- 2018

A flow method for the dual Orlicz–Minkowski problem

- Mathematics
- 2020

In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a…

Noncompact L_p-Minkowski problems

- Mathematics
- 2018

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability…

## References

SHOWING 1-10 OF 31 REFERENCES

On the _{}-Minkowski problem

- Mathematics
- 2003

A volume-normalized formulation of the Lp-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all p > 1, including the degenerate case where the index p…

The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem

- Mathematics
- 1993

The Brunn-Minkowski theory is the heart of quantitative convexity. It had its origins in Minkowski's joining his notion of mixed volumes with the Brunn-Minkowski inequality. One of Minkowski's major…

Classification of limiting shapes for isotropic curve flows

- Mathematics
- 2002

with a 5 0, and initial condition x(p, 0) = xzo(p). This produces a family of curves yt = x(Si, t). Here n is the curvature, and n is the outward-pointing unit normal vector. These equations are…

Motion of hypersurfaces by Gauss curvature

- Mathematics
- 2000

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in…

Sharp Affine LP Sobolev Inequalities

- Mathematics
- 2002

A sharp affine Lp Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp Lp Sobolev…

Self-similar solutions for the anisotropic affine curve shortening problem

- Mathematics
- 2001

Abstract. Similarity between the roles of the group $SL(2,\bf R)$ on the equation for self-similar solutions of the anisotropic affine curve shortening problem and of the conformal group of $S^2$ on…

Partial Differential Equations Invariant under Conformal or Projective Transformations

- Mathematics
- 1974

A variational theory of the Hessian equation

- Mathematics
- 2001

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of…

A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity

- Mathematics
- 1990

The purpose of this note is to show a localization property of convex viscosity solutions to the Monge-Ampere inequality 0 1−(2/n)) are strictly convex