We consider the problem of camouflaging for bodies with specular surface in the framework of geometric optics. The index of visibility introduced in [Plakhov 2017] measures the mean deviation of light rays incident on the body's surface. We study the problem of minimal visibility index for bodies with fixed volume contained in the unit sphere. This problem is reduced to a special vector-valued problem of optimal mass transfer, which is solved partly analytically and partly numerically. This paper is a continuation of the study started in [Plakhov 2009], [Plakhov 2017], and [Plakhov and Roshchina 2011].