Isoperimetry and volume preserving stability in real projective spaces

We classify the volume preserving stable hypersurfaces in the real projective space ${\mathbb{R}\mathbb{P}}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $$\mathbb{R}\mathbb{P}$^k\subset $\mathbb{R}\mathbb{P}$^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritor'{e} and A. Ros on ${\mathbb{R}\mathbb{P}}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in ${\mathbb{S}}^n$.