Quantitative stability of the intersection body operator near the ball, and the dynamical origin of the two--dimensional degeneracy

Let $\text{IB}$ denote the intersection body operator on star bodies in ${\mathbb{R}}^n$. A recent theorem of Milman, Shabelman and Yehudayoff establishes that for $n\ge 3$ the equation ${\text{IB}}^{2}K=cK$ holds if and only if $K$ is a centered ellipsoid, thereby resolving the fixed–point problem for ${\text{IB}}^2$ and, as a consequence, the long–standing conjecture $\text{IB}K=cK\iff K$ is a ball. We complement this qualitative rigidity with a \emph{quantitative} analysis in a neighbourhood of the ball. Linearizing the associated shape dynamics on $L^2(\text{Sph})$, we compute the full spectrum of the operator ${\text{IB}}^2$ at the ball in closed form for every dimension: the degree–two (ellipsoidal) harmonics are neutral with multiplier exactly $1$, while all higher harmonics are contracted, with a sharp spectral gap $$\mathrm{g}\mathrm{a}\mathrm{p}(n)=\frac{(n-2)(n+4)}{(n+1{)}^2}\mathrm{.}$$ This yields an explicit linear stability constant $C(n)=(n+1{)}^2\mathrm{/}((n-2)(n+4))$, and, via a center–manifold reduction, a local quantitative stability statement for ${\text{IB}}^2$ near the ball valid in each fixed dimension $n\ge 3$. The gap degenerates precisely as $n\to 2^{+}$, giving a transparent \emph{dynamical} explanation of the well–known exceptional status of the plane, where $\text{IB}K=2K$ for every origin–symmetric star body. We also record the reduced normal form of $\text{IB}$ on the ellipsoidal directions and observe that the centered ellipsoids constitute a normally attracting invariant manifold for the shape under iterated intersection bodies. The methods are perturbative and do not address the global periodic problem ${\text{IB}}^{m}K=cK$ for $m\ge 3$, which we discuss.