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Let $ D$ be a bounded open subset of $\mathbb R^n$ with $|\partial D| <\infty$ and let $x_0 $ be a point of $D$. We introduce a new parameter, that wecall Kuran gap of $\partial D$ w.r.t. $x_0$. Roughly speaking, this parameter,denoted by $\mathcal{K}(\partial D, x_0)$, measures the gap between $u(x_0)$and the average of $u$ on $\partial D$ for a particular family of functions $u$harmonic in $\overline{D}$, in terms of the Poisson kernel of the biggest ball$B$ centered at $x_0$ and contained in $D$. To do that, we need the domain $D$Lyapunov-Dini regular in at least one of the points of $\partial D$ nearest to$x_0$. Our main stability result can be described as follows:$\mathcal{K}(\partial D, x_0)$ is bounded from below by a kind of isoperimetricindex, precisely the normalized difference beetween $|\partial D|$ and$|\partial B|$. This extends a stability result by Preiss and Toro, and a morerecent theorem by Agostiniani and Magnanini. Moreover, from our stabilityinequality we obtain a new sufficient condition for a harmonic pseudosphere tobe a Euclidean sphere, a result which partially improves a rigidity theorem byLewis and Vogel. Finally, we give a new solution of the surface version of asolid ``potato'' problem by Aharonov, Schiffer and Zalcman.

On the harmonic characterization of the spheres: a sharp stability inequality and some of its consequences