Prescribing capacitary curvature measures on planar convex domains (Q1985348)

Let \(\mu\) be a finite non-negative Borel measure on the one-dimensional unit sphere. The author considers the question if there exists a bounded non-empty open convex set \(\Omega \subset \mathbb{R}^2\) such that \(d\mu_p(\overline{\Omega}, \cdot) = d\mu(\cdot)\). He shows that the answer is positive if and only if \(\mu\) has the centroid at the origin and its support \(\mathrm{supp}(\mu)\) does not comprise any pair of antipodal points.