The planar \(L_p\) dual Minkowski problem
The Minkowski-problem asks for the construction of a convex compact surface whose Gauss curvature is given. It is connected with the famous Brunn-Minkowski-Inequality. Of course other measures of curvature may be given, e.g. the \textit{dual curvature measures}. This family of measures connects the cone-volume measure and Aleksandrov's integral curvature. The associated Minkowski problems are called dual Minkowski problems. The goal is to solve a given nonlinear differential equation, depending on parameter \(p\) and its dual \(q\), for a given function \(f\) defined on \(S^{n-1}\). For the planar case \(n = 2\) solutions for many cases in \(p\) and \(q\) are already known. The authors present in the paper a solution for the (unsolved) case \(p < 0 < q\). In particular they prove that for any positive smooth function \(f\) on \(S^1\), there exists an function \(F\) with the property that if \(F(q) < p < 0\) or \(0 < q < -F(-p)\) there is a solution to the planar \(L_p\)-dual Minkowski problem. The requested function \(F\) depends on \(p\) and \(q\) only and is given in detail. John ellipsoids are used to find bounds on the width of the requested body and these bounds lead to the prove that a the requested solution exists.
scientific article; zbMATH DE number 7369819
scientific article