[en] Spherically symmetric anisotropic static compact solutions to the Einstein equations in dimension $d⩾<!--\; ???\; -->4$ are considered. Various matter models are examined and upper bounds on the ratio of the gravitational mass to the radius in these different models are obtained, and saturation of these bounds are proven. Bounds are also generalized in the presence of a non-zero charge and a positive cosmological constant. These bounds are then used to find the maximum of the gravitational redshift at the surface of the object. (paper)