Isoperimetry and volume preserving stability in real projective spaces

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (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 $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.