In this talk, I will begin with a brief review of some foundational geometric inequalities for hypersurfaces in Euclidean spaces focusing on those where equality charaacterizes the standared geodesic spheres. A prime example is Reilly's celebraated inequality which provides a sharp upper bound for the first non trivial eigenvalue of the Laplace Beltrami-operator on compact, embedded hypersurfaces. Interestingly, these inequalities can often be traced back to a fundamental estimate involving the $L^2$ norm of the position vector. I will then delve into a novel stability refinement of this inequality: we establish that if a hypersurface nearly attains this lower bound on the $L^2$-norm, then it must be geometrically close, in a suitable sense, to a round sphere.
