On the planar L-Minkowski problem

ON THE Lp-MINKOWSKI PROBLEM

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 is equal to the dimension of the ambient space. A new approach to the Lp-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all p ≥ 1. The Minkowski pr...

THE L p MINKOWSKI PROBLEM FOR POLYTOPES FOR p < 0

Existence of solutions to the Lp Minkowski problem is proved for all p < 0. For the critical case of p = −n, which is known as the centro-affine Minkowski problem, this paper contains the main result in [71] as a special case.

Optimal Sobolev Norms and the L Minkowski Problem

The Logarithmic Minkowski Problem

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

On the Lp Minkowski Problem for Polytopes

Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1. As observed by Schneider [21], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R, with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classica...