  Alexandrov's Soap Bubble theorem dates back to $1958$ and states that a
compact embedded hypersurface in $\mathbb{R}^N$ with constant mean curvature
must be a sphere. For its proof, A.D. Alexandrov invented his reflection
priciple. In $1982$, R. Reilly gave an alternative proof, based on integral
identities and inequalities, connected with the torsional rigidity of a bar. In
this article we study the stability of the spherical symmetry: the question is
how much a hypersurface is near to a sphere, when its mean curvature is near to
a constant in some norm. We present a stability estimate that states that a
compact hypersurface $\Gamma\subset\mathbb{R}^N$ can be contained in a
spherical annulus whose interior and exterior radii, say $\rho_i$ and $\rho_e$,
satisfy the inequality $$ \rho_e - \rho_i \le C \Vert H - H_0
\Vert^{\tau_N}_{L^1 (\Gamma)}, $$ where $\tau_N=1/2$ if $N=2, 3$, and
$\tau_N=1/(N+2)$ if $N\ge 4$. Here, $H$ is the mean curvature of $\Gamma$,
$H_0$ is some reference constant and $C$ is a constant that depends on some
geometrical and spectral parameters associated with $\Gamma$. This estimate
improves previous results in the literature under various aspects. We also
present similar estimates for some related overdetermined problems.
