The Lp Minkowski problem for negative p

Determine necessary and sufficient conditions for solvability of the L_p Minkowski problem for all negative parameters p<0, thereby resolving the cases that remain unresolved beyond the established partial existence results.

Background

The paper places its discrete existence results within the broader L_p Minkowski problem, which prescribes the L_p surface area measure of a convex body. Although substantial results are known for various parameter ranges and classes of measures, the authors explicitly state that the negative-parameter regime remains largely unresolved. The paper addresses a discrete subclass but does not claim to settle the general problem for p<0.