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On the planar Gaussian-Minkowski problem
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The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this ``uniqueness'' result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.
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Cited by 2 Pith papers
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Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
For convex bodies in R^3, bounded L_p qth dual curvature with p in [0,1) and q>2+p forces a uniform diameter upper bound and volume lower bound.
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Uniqueness in the near isotropic Lp dual Minkowski problem
For -1<p<1 and q sufficiently close to n, the near-isotropic Lp dual Minkowski problem on the sphere has a unique solution, with a sharp C0 estimate; the even case covers -1<p<q<min{n,n+p}.
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