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In this paper we study the solvability of the rotationally symmetric centroaffine Minkowski problem. By delicate blow-up analyses, we remove a technical condition in the existence result obtained by Lu and Wang [30].

In this paper we show by example that there is no uniform estimate for the L p-Minkowski problem. As a result we obtain the nonuniqueness of solutions to the problem.

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi–Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi–Yau category is simply connected in the sequel [King and Qiu, Invent. Math., to appear].