Prescribing Capacitary Curvature Measures on Planar Convex Domains

For p is an element of(1,2]and a bounded, convex, nonempty, open set omega subset of R2 let mu p(omega over bar ,center dot)be the p-capacitary curvature measure (generated by the closure omega over bar of omega on the unit circle S1. This paper shows that such a problem of prescribing mu pon a planar convex domain: "Given a finite, nonnegative, Borel measure mu on S1 find a bounded, convex, nonempty, open set omega subset of R2such that d mu p(omega over bar ,center dot)=d mu(center dot) is solvable if and only if mu has centroid at the origin and its support supp(mu does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if d mu p(omega over bar ,center dot)=psi(center dot)dl(center dot) with psi is an element of Ck,alpha and dlbeing the standard arc-length element on S1 then partial differential omega is of Ck+2,alpha