We propose a new type of axisymmetric dielectric target which effectively concentrates Cherenkov radiation (CR) generated in the bulk of the material into a small vicinity of a focus point. It can be called the "axicon-based concentrator for CR."A theoretical investigation of radiation field produced by a charge moving through the discussed radiator is performed for the general case where a charge trajectory is shifted with respect to the structure axis. The idea of a dielectric target with a specific profile of the outer surface was presented and developed in our preceding papers. However, contrary to the previous configuration of such a target (which was investigated for both centered and shifted charge trajectory), the current version of the device allows the efficient concentration of CR energy from relativistic particles, making this device extremely prospective for various applications.