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

Abstract

Let $\IB$ denote the intersection body operator on star bodies in $\R^n$. A recent theorem of Milman, Shabelman and Yehudayoff establishes that for $n\ge 3$ the equation $\IB^2 K = cK$ holds if and only if $K$ is a centered ellipsoid, thereby resolving the fixed--point problem for $\IB^2$ and, as a consequence, the long--standing conjecture $\IB K = cK \Leftrightarrow K$ 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 $L^2(\Sph)$, we compute the full spectrum of the operator $\IB^2$ 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 \[ \mathrm{gap}(n)\;=\;\frac{(n-2)(n+4)}{(n+1)^2}. \] This yields an explicit linear stability constant $C(n)=(n+1)^2/\big((n-2)(n+4)\big)$, and, via a center--manifold reduction, a local quantitative stability statement for $\IB^2$ 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 $\IB K = 2K$ for every origin--symmetric star body. We also record the reduced normal form of $\IB$ 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 $\IB^m K = cK$ for $m\ge 3$, which we discuss.