The convex back-scattering support

A monochromatic, i.e., fixed-frequency, back-scattering kernel measured at all angles does not uniquely determine the index of refraction in an inhomogeneous medium, nor can it guarantee any upper bound on the support of the inhomogeneity. We show that it is possible to associate with any such kernel its convex back-scattering support, a convex set which must be a subset of the convex hull of the support of any inhomogeneity with that back-scattering kernel. For the Born approximation, we further demonstrate that there is an inhomogeneity supported in any neighborhood of the convex back-scattering support which has exactly that back-scattering kernel. Last, we discuss a practical implementation of these results and include a numerical example.