Diameter estimate for planar 𝐿_{𝑝} dual Minkowski problem

Abstract

In this paper, given a prescribed measure on S 1 \mathbb {S}^1 whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar L p L_p dual Minkowski problem when 0 > p > 1 0>p>1 and q ≥ 2 q\ge 2 . We also prove the uniqueness and positivity of solutions to the L p L_p Minkowski problem when the density of the measure is sufficiently close to a constant in C α C^\alpha .