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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

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    $\begingroup$ Yes, you should be able to construct such a metric. Form "bubbles" on $R^2$ i.e. spheres connected sum with the flat plane to produce a small neck. It's clear that the neck is very good isoperimetrically. Now, make two spheres that contain volume V together, and have total neck area A, and make another single sphere with volume V and area A. You should be able to argue that the isoperimtric problem at this volume V has a solution given by two disks and also one given by one. It may take a bit of work to make this rigorous, but I am pretty sure it can be done by a limit argument.. $\endgroup$
    – Otis Chodosh
    Commented Aug 23 at 23:51
  • $\begingroup$ @OtisChodosh Thank you very much for your attention. I would appreciate if you give a complete proof. $\endgroup$
    – Ali Taghavi
    Commented Aug 23 at 23:58

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