Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:

$$\ell(M)=\ell(N)=\ell, A(M^o)=A(N^o)=A$$

and for every compact hypersurface $P$ with $\ell(P)=\ell$ we have $A(P^o)\leq A$

Here $\ell(M)=\int_M vol_M$ the natural volum form on $M$ naturally arises from the volum form on the underling manifold. $A(M^o)$ is the area of the bounded component of $M^c$ wrt the volum form arising from the metric

Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:

$$\ell(M)=\ell(N)=\ell,\;\; A(M^o)=A(N^o)=A$$

and for every compact hypersurface $P$ with $\ell(P)=\ell$ we have $A(P^o)\leq A$

Here $\ell(M)=\int_M vol_M$ the natural volum form on $M$ naturally arises from the volum form on the underling manifold. $A(M^o)$ is the area of the bounded component of $M^c$ wrt the volum form arising from the metric