Polymer-induced interactions between two small mesoscopic particles of radius R and between a particle and a wall are investigated for dilute or semidilute embedding solutions of long flexible nonadsorbing polymer chains with radius of gyration R-g. Asymptotically exact predictions are obtained using a "small radius expansion," to express the interactions in terms of properties of the polymer solution without particles. The nonmonotonic dependence of the second virial coefficient B-2(CC) of a dilute suspension of particles on the interchain overlap is discussed in detail. The magnitude of the minimum of B-2(CC)/R-3 increases as a power law in R-g/R. The exponent and amplitude are quite different from the earlier prediction of an integral-equation approach. For dilute polymers in two dimensions outside two circular disks in contact, exact results are given for the monomer-density depletion profile, the pressure variation along the perimeter of, and the polymer-induced force between the two disks.