Although the recent complete solution of the Willmore conjecture depends on the min-max theory of minimal surfaces, some partial results can be obtained by using the isoperimetric problem on 3-manifolds of positive Ricci curvature and, in particular, on the real projective space. Using this tool, we will present a proof of the Willmore conjecture for antipodal invariant tori in the 3-sphere and for the case of tori with a central symmetry in Euclidean 3-space.