Let D be a convex body in R^n. The Gauss map g: /partial D -> S^{n-1} is determined by g(x)=v, the outer unit normal at x in /partial D. The classic Minkowski problem asks for which measure dm on (S^{n-1},ds) one can find a convex body D such that g_*(ds)=dm. This talk is to discuss the similar problem with a measure induced by the logarithmic capacity in place of surface measure.