An absorbing body is assumed to absorb the fraction α of any radiation falling on it, independent of the angle of incidence, and to reflect the remaining fraction, 1 − α, diffusely. The body is partly surrounded by a perfectly (but not necessarily specularly) reflecting cavity. An aperture admits uniform fully diffuse radiation to the cavity. The whole system is assumed to be cylindrical, i.e. to be invariant under translation in the direction of a cylinder axis. An upper bound on the efficiency of absorption, previously obtained by O'Gallagher and others, is confirmed by another argument. The upper bound is α/(α+ (1 − α)C) whereCis the ratio of aperture to absorber area.